Calculus for beginners - Limits of Functions | Dániel Csíkos | Skillshare

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Calculus for beginners - Limits of Functions

teacher avatar Dániel Csíkos, Mechanical engineer

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Taught by industry leaders & working professionals
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Lessons in This Class

26 Lessons (1h 55m)
    • 1. Welcome to the course!

    • 2. Functions

    • 3. Definition of limits

    • 4. Special limits

    • 5. Fundamental operations

    • 6. Limits of fractions – 1. Example

    • 7. Limits of fractions – 2. Example

    • 8. Limits of fractions – 3. Example

    • 9. Getting rid of square roots – 1. Example

    • 10. Getting rid of square roots – 2. Example

    • 11. Classification of discontinuities

    • 12. Investigation of discontinuities – 1. Example

    • 13. Investigation of discontinuities – 2. Example

    • 14. Investigation of discontinuities – 3. Example

    • 15. Trigonometric functions – 1. Example

    • 16. Trigonometric functions – 2. Example

    • 17. Trigonometric functions – 3. Example

    • 18. Limits based on Euler's number (e) – 1. Example

    • 19. Limits based on Euler's number (e) – 2. Example

    • 20. Limits based on Euler's number (e) – 3. Example

    • 21. L’Hospital rule

    • 22. L’Hospital rule – 1. Example

    • 23. L’Hospital rule – 2. Example

    • 24. L’Hospital rule – 3. Example

    • 25. L’Hospital rule – 4. Example

    • 26. Thank you!

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About This Class

This mathematics course has mainly been created for students currently learning calculus at college/university. In this course, you are going to find everything that you need to know about the limits of functions!

Objective of the Course

The main objective of the course is to help you be able to solve any kind of problems related to the limits of functions.

What will I learn?

  • How to calculate the limit of functions
  • Limits of fractions
  • Classification of discontinuities
  • How to deal with square roots
  • How to calculate the limits of trigonometric functions
  • How to use Euler's number (e) to determine limits
  • L'Hospital rule
  • Additional know-how and useful tricks

What do I need to know to start the course?

A basic pre-calculus knowledge is enough. The course starts from scratch and take you through the topics with lots of examples.

How to make the most of this course?

There are several practice problems that you can solve by yourself. I suggest solving those problems after watching the lectures of each topic!

Meet Your Teacher

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Dániel Csíkos

Mechanical engineer


Welcome! :)

I'm a mechanical engineer and online entrepreneur. I've got my MSc level degree in mechanical engineering at Budapest University of Technology and Economics and I'm enthusiastic about sharing my knowledge and my love of engineering sciences.

I've been teaching and helping mechanical engineering students as a private tutor in various subjects for over 3 years. Therefore I not only know how to understand a topic as a student but also how to make it understandable for others. I passed most of my subjects with an excellent mark and I hope I can help you to reach the desired mark or objective for yourself by ease, too. I offer guidance to understanding either if you are a student or if you want to... See full profile

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1. Welcome to the course!: Welcome to my course. Let me introduce myself and my course about the limits off functions just in the natural. I am Daniel Chico. Sh I'm a mechanical engineer, but my main focus is on teaching you and others online. I've been in an extraordinary mathematics vacuity in secondary school. I cannot say that I was the best, but I've experienced the best training available. This inspired me to help you. I don't want to let you struggle with mathematics. It is a fine topic, but unfortunately, it is not always tilt. Well, I have several years of teaching experience and I can put myself into your position. So I think I'm the van Who can help you? I designed the scores for students having trouble with calculating the limits of functions . I tell you that you're in a nutshell. As I don't want to concentrate on solving problems. Therefore, I show you lots of numerical examples as I believe that this is the better way of learning any topic. But it is never enough to just see the solution. I would like you to get involved. You are going to get practice problems designed for you. These are going to help to really understand the topic by applying everything you've learned. So what is going to be in the course? A short, heretical introduction about the basics? Nothing to matter. Month ago them, we jumped deep into calculating limits. First, you can learn how to calculate the limits of fractions. Then I show you how to handle square roots in limits on then other applications. With some good knowledge at your hand, we jump into the classifications off these continuities. After that, you are going to become an expert of the limits off big automatic functions on the limits based on me. And finally I introduced the Lopa Teruel, which is the with you mate, to work to determine trouble. Some limits. This is all you need if you want to determine limits of functions, so join me and let's learn everything in practice 2. Functions: before calculating the limits off functions. Let me tell you a bit about the functions and their limits in general. First the fall. But functions are a function is a relation for which amount off the domain corresponds to exactly one amount off the range. For example, F X equals two X squared. But X is the element of the set off rheal numbers. To get the correct definition, the domain should also be mind. This is why I've right that X is the element off the set off Rio numbers, which is indicated by the symbol similar to power. This function relationship means that for every extra number, the function gives your results, which exhausted ax squared. In this example, there is a result for the function in case of any point of the domain. Let me highlight the compared to the sequences. The function is not necessarily discreet valued. It can also be continuous. I don't only mean that the results of the function ist continuance, but also its domain can be continues to in case of sequences that are just lifts off numbers with the given sacrificial number. This is a huge difference between of sequences and functions. But besides this, if you know how to calculate limits off sick Frances, you can use it in case of functions. Also, if you learn how to calculate the limits of functions, that knowledge is useful in case of sequences do. However, the functions can be differentiated, integrated, and so there are more operations that you can do with them. These cannot be applied to sequences. So, for example, the low Peter will only stands for continuous functions. Let's go ahead and learn how to calculate the limits of functions. 3. Definition of limits: Let's learn, but the limit actually is. We can start with the notations. Limb F AXB equals the capital. Al vile eggs goes to a Is the limit calculation unusual format Here l is the limit off the function. F acts around the point where X equals two A. The limb means Lee mask, which is a Latin word that basically means limit. Any function can be investigated around any point around which the points are within the domain of the function. But the domain of the function doesn't have to contain point A. So A doesn't have to be in the domain, but the points around it should be in the domain of the faction, So basically the limit can be calculated at any point or even at class for minus infinity. But when does a limit exists on, but it's it value there exists limit. If they're inequality. Absolute value. F x minus. L smaller than ab silent is satisfied whenever the absolute value of X minus A is smaller than delta. Okay, but what does this mean? Let's start with the second inequality. Eight marks a number on the X axis around which the limit shall be determined if the difference off X and A is smaller than Delta than the points are taken from the small neighborhood off A. The function gives a result for every point around. A. If the difference of these function values on the limit is smaller than given up silent, then we can say that for X values around a, the function takes values around capital, L said. That becomes the limit for a given up silent. We can calculate the sufficient value off data. Therefore, AB silent is arbitrary, and that is just something that we can calculate. If there is a limit of silent can be choosing as small as you wish. The function values are going to get close to the limit and embrace, and we can even calculated out of forgiven Upsilon. Let me put this into my own. No mathematical birds. There is a limit if the function values are basically the same at a given short, the main around the point. I hope that the meaning off limit is clear, more or less, However, you don't even have to understand. You are going to see the main points during the numerical examples. Now let's just go through the main types off limits. There is the basic case where we check the limit around the given value off acts. This is what we have marked with a. So in this case, X goes to a. We can also check the limit at plus T infinity or at minus infinity. These are a bit different in theory from the original case, but you can basically the every damn in the same day. Also, there are one sided limits. There's a limit from the right or so called limit from above. In this case, acts is slightly greater than a and you change the limit with this condition only for the X values greater than any in the other. Case X is smaller than a. It is called the limit from the left or the limit from below. Here, the X values are slightly smaller than a, and they are approaching A from below this five options can occur in practice, and you are going to see how you can handle them. 4. Special limits: Let's go through the special limits. These are advanced that you can be a don't in the later examples. There are some special limits at infinity, but they are quite obvious. The limit off acts is infinity as X goes to infinity or so X to the power of K goes to infinity. If K is greater than zero, so either you face with over term or with the root of facts. One over X goes to zero at infinity as it could be written as one over infinity. This is actually acts the power off minus one so you can see that the negative powers off acts are going to zero infraction format. One over X to the power of K goes to zero If Gaius Positive. Let's see the accident and short terms A to the power of acts goes to infinity. If the base of the exponential terms is greater down bomb, for example, you had X or two X is divergent. They are going to infinity if the base of the exponential terms is smoother down than a at Axe Ghost zero, for example, 1/2 to the power of acts goes to zero as by every multiplication. That expression gets smaller and smaller. I skipped a vote, won. Its limit is simply one, as it becomes constant. One for any axe. Let's see somewhere interesting limits. The limit off extra route see is Monnet infinity, where C is an arbitrary, really constant or so the limit off extrude X is one of infinity. These are good to know. If you know them, you can easily saw off. Limits are extra. Routes are included. The limit off one plus one over X to the power of acts becomes E. This is a result that is rather famous for sequences as the definition off Eilers number was done according to the sequence version of this limit, However, this is also true for functions. Finally, that may show you limit, but we are not working at infinity. The limit off sine X over X is one as X approaches zero. This is a very useful result. You must use this every time when you are working with three Ghanem Africa terms around zero. Also, this can be very useful. Generally, if you perfectly grammatical functions, you're going to see the use of this on much more during the sample problems 5. Fundamental operations: let me show you the fundamental operations. Andi. Even more importantly, their concept France's, which you can use in practice as a starting point. We know that the limit off F ax is capital A and the limit of G ax is capital B as X approaches. A. This A can be anything now either rial number or even infinity minus infinity, or we can calculate the van sided limit. If we take the summer off the functions and check the limit, we find that the limit off the sum is the same as some off the limits off F ax on G X. Similarly, in case off the difference off the functions, we can calculate the resultant limit by just taking the difference off the limits off half acts on GXE If a function F X is multiplied by a really constant, see the limit off see Times X becomes C times the limit off acts. The same goes for the multiplication off the functions. The limit off affects stance. G X becomes capital. A Times Capital B Capital eight is the limit of effects on capital. B is the limit of GXE. What's so the oceans can be calculated in a similar manner. The limit off half acts over G ax equals two capital A over capital B. Of course, G X and B cannot equal to zero as we can be ride by Sierra. Let's continue with the consequences off this previous lows. If the limit off F acts is infinity and it is multiplied by a non cereal constant, the limit of the product is also infinity. This is obvious if the limit off F X is zero on G X is not divergent, so it's limit is not infinite your minus infinity than the limit off their productive zero . If you multiply zero with infinite number, the result is also zero. This is a natural concept fans. Similarly, if the limit of F axes infinity on the limit of GX is not zero but a positive number, then the limit of the product is also infinity. If an infinite number is more deployed by a non cereal farm than the result is infinite, to if the limit off G axis negative than the limit of the product would become minus infinity simply, the signs are taken into account. We can continue with the quotations if the limit of F X is infinity and the limit of GX is a positive finite number. The limit off halfbacks over G tax becomes infinity as an infinite number is derided by something that is not infinite. The result is still on infinite Mom. The positivity is only needed to have plus infinity. As a result, if the limit off GX is negative than the result is minus infinity. If the limit off after tax is a positive number, even finite or infinite than the limit of G X is zero then the limit off affects over GX is infinity. Technically, you couldn't divide by zero as it is stated in the law. However, in practice we use this formula even for division with zero here there are two important respects. First of all, you do not actually divide with zero in the courts and you divide with something that goes to zero. But the denominator is never cereal as a number is approaching zero, it's reciprocal approaches infinity. That is why we go this result. On the other hand, you should be careful with the science. This result is only valid if G X is approaching zero from above. In this way, we have a positive result on the limit. This plus infinity. If g x can be positive a negative to that we don't have a valid limit. Don't worry. If this is trading, I'm going to clear this for you later when we are dealing with examples. Finally, if the limit off F X is a finite number on the limit of G axis infinitive than the limit off affects over GX zero, it doesn't matter what is the sign off effects. If a finite number is divided by an infinite involved than the result just goes to zero. These consequences are pretty useful in practice. But really, if they are not clear yet, don't worry. Just stay with me for the examples. 6. Limits of fractions – 1. Example: Let's start the sample problems with an important special case, we should determine the limit off three X squared plus 4/6 X squared plus nine x as X approaches infinity as ex Gusti and Phanatic. So that's X crowd that for both the nominator and the denominator. Go to infinity. But you don't have to worry about it in case of investigating fractions at Infinity, there's a basic trick. If you divide every term off a fraction with the greatest term, you can easily see what the limit is. What is the greatest term now, the one which would be the greatest at infinity? This means the greatest power off acts. Therefore, the greatest term is X squared. In the present case, let's divide every term. By that, I indicate the division by writing X squared over X. Clad in front of the fraction, this is just a multiplication by from the new nominator is three X squared plus four over X squad, which is equal to three plus four over X scrapped. The new denominator is six X squared plus nine x over X quad, which is equal to six plus nine over acts. Just let me remark that we could have divide X as it doesn't eat for 20 as we are investigating the problem. As X goes to infinity, we can't skip the X equals zero case. So back to a problem after the division. By the greatest term, you can check the limit off each expression in the nominator and in the denominator as X and X squared are going to infinity. Four over X crowd on nine over. Acts are going to zero This coach ins I mean a finite number divided by an infinite one for which the limit is always zero. Therefore, only treat remains in the nominator on six. In the denominator, the limited hole fraction becomes vulnerable to this is the mattered. If X goes to infinity, you can always work like this. 7. Limits of fractions – 2. Example: Now let me show you an example in which the limit is investigated around the rial number and not that infinity. We should determine the limit off X squared minus one over X minus one. As X approaches from first, you should always simply check the nominator on the denominator. If X goes to bomb, X crowd goes to one squad, which is vom. Therefore, denominator goes toe one minus one budget was 20 Similarly, the denominator also goes toe one minus one as X approaches Mom, as the limits off the differences are zero, we end up with a fraction going to 0/0. This is something that we cannot handle directly. If you divide with a number approaching 20 the result would tend to infinity. However, the results with 10 to 0 Because of the zero in the nominator, you should rearrange this fraction somehow in order to avoid the zero over Sierra format, Just try to think of the basic knowledge about Paulina meals. X squared minus one can be written as a product X squared minus one equals two X plus one times X minus one. This is a result that comes from a special identity. However, if this is not an obvious thing for you, I have a basic trick that you can always apply if the nominator and the denominator is zero at a given point. That point is basically a route of the immigrations X squared minus one in 40 and X minus Warning for zero. So X equals toe. One is a root. Why is it good for us? If you know the route, you know that X minus. That route is going toe appear in the factored form of the polynomial. This factor form means the form of the bullying, a meal where you fried the point meal up by a product off Pelino meals with smoother degree . You can be sure that X minus one appears in the factor form off X squared minus one. After that, you can guess the function that you should use as the multiplier, as it is also going to be explosive. Something you have a nascent job. X Times X gives you X scrap, then minus one times something equals two minus one. What is this? Something it should be plus one or so You should check the remaining terms plus one times X is X on minus one times X is minus sex. There some is zero, which is good as X is not in the nominator. There's only X squared Anyways. You can just do this, factoring at all times. After you have got this form, you can simply fight by X minus one. It is important to know that we hasn't divided by zero X minus bond goes to zero, but it can never equal to zero, as this fraction is just not valid at X equals to one as its original denominator can never be zero. Therefore, X equals one is not in the domain of the function, so we don't divide with zero. Now we only need to check the limit off X plus one. It is, too, as X approaches one. But this we have reached the solution. You can always use this matter in case of calculating the fractions off Pelino meals. If 0/0 case appears, you can always leave. Ride the Polina meals in factored format and you can simplify with the problematic terms. Finally, something simple remains for which you can easily calculate the limit 8. Limits of fractions – 3. Example: in this lecture, I show you how to solve a complicated problem. We should determine the limit off X plus two over square root eight X squared plus four plus three x as X approaches minus. Do before you try any kind off reformulation, just check the limit. Maybe it can easily be calculated directly X goes to minus two, according to the condition that four X goes to minus two in the nominator, so the nominator goes to minus two plus two, which is zero. Let's check the denominator three X goes to minus six as X approaches minus do X squared goes to manage to scratch, which means plus four. Therefore, eight times X crab goes to 32 4 should be added to this 32 so the limit off the term under the route is 36. The square root of 36 is six. That for the first part of the denominator goes toe plus six. As the second part of the denominator goes to minus six, the some of them goes to zero. So the fraction is in a zero over serial form. We must reformulate to get a valued result. We can't really do anything about the nominator, so we should try to deal with the denominator. What can we do if you see square roots? There is a basic matter to apply. There's a violent identity is quiet, minus B squared equals two a minus. B times a plus B. This identity is used whenever you have to de of its square roots, and you face any kind of troubles. By using this formula, we can get rid of the square root of the denominator. A plus B can be related to square root eight X squared plus four plus tree X. If we multiply with a minus bi, the square root disappears, but you cannot just multiply the denominator to have exactly the same expression as the origin of um, you should multiply and also divide with the same expression so we multiply and divide the whole original expression by square root. Eight X squared plus four minus three x. Then we can take the product of the denominators according to the identity. The result is in the form of a squad minus B squad as square root eight X squared plus four took the roll off A. Its squad is eight x squared plus four. Three x to the role of Be So it's scratch is nine x crowd and it should be subtracted off course. You can just take product on get the same result in the denominator. Why is this stop useful for us now? We can just calculate the difference in the denominator which might result in a term that doesn't go to zero that is or hope. After the Division four miners, X crowd remains in the denominator. Now we can check the limit again. Well, it turns out that we are not really as X goes to minus two, X squared goes to plus four four minus four goes to zero. So the denominator goes to zero. Also, X plus two goes to zero as X approaches minus two. So we are still dealing with 0/0 form. But don't worry or works has not been useless in this form. We can just go on reformulating according to the basic trick known for polynomial fractions , the square root doesn't really matter. Be I only have a problem with X plus 2/4 minus X crab. In this case, we can get the pulling no meals into factor for so let's get the denominator into factor for is X equals two. Minus two is a root of four minus X. Craddick lost zero that is going to disappear as X plus two. The other factor is going to be two minus x as two minus X times two plus X equals 24 minus X crab. Now the X Plus two has appeared boost in the nominator and in the denominator, we can just simply five it it. That's how we get a new faction. We should investigate. Square root eight X squared plus four minus three x over two minus x as X approaches minus two. Now we are going to get a result. Let's check this limit term by term. X quad goes to minus two Squared, which is for three times X goes to three times minus two, which is minus six on the axe in, the denominator goes to minus two. Altogether, the nominator goes to 12 on, the denominator goes to four, so the whole a fraction goes to three. This has been quite a challenge, but if you see square root, you can always use the first trick that I showed you. And if you see fractions off Pelino meals, you can use the second trick at the end, you are certainly going to have a result. 9. Getting rid of square roots – 1. Example: I have already shown you a trick. How to handle square roots. You can always use the same mattered in case of trouble. Some limits with square root. Let me show it to you. Through an example, we must determine the limit off squarer to explosive on minus square. Root two X minus. Four. As X goes to infinity, the first term goes to infinity as X goes to infinity. Taking the route only slows the growth of the function, but the result is still going to be infinity at infinity. The same goes with the second term. These are obvious, but what's the limited difference? We cannot calculate the limit off infinity minus infinity, at least not directly. We need a trick. There is a Valium formula to use. A squad minus B squared equals two a minus. B times a plus B. Either we subtract or add to square roots so the original four month can be identified as a minus bi or a plus B. If we do a multiplication, we get the result that the square terms appear that for you get rid of the square root and we can subtract the terms under the square roots. Usually this gives you the final solution. This time, a minus bi equals to the function that we have to investigate. If we multiplied this bit a plus B, we get rid of the square roads. However, we cannot do a multiplication by itself as it would change the limit being multiply and divide with the same term. Therefore, the investigated function does not change this time we multiply and divide by square root. Two x plus one plus square root two x minus four. The multiplication gives the result in the four months off a squad minus B squad in this manner because Rudolph the original squirrels, therefore the nominator doesn't contain on the square root off course, we have made a fraction and the denominator contains square root. But it is not going to be a problem. Now we can do the subtraction in the nominator two x minus two X drops out. So only von minus minus four remains which exists to five. Now we can check the limit. Vance again, the nominator obviously goes to five in the denominator. Both terms go to infinity as X goes to infinity, just like in case of the original function. But this time we do not have to calculate the difference. We haven't addition to deorbit infinity, plus infinity is infinity. So the denominator goes to infinity as we divide a finite number. But the number going to infinitely the result tends to zero That, for the limit is zero. In this case, if you ever see square roots and have a problem with the limit, tried this matter. 10. Getting rid of square roots – 2. Example: let me show you another example of its square root. We must determine the limit off square root, X clad plus x plus one minus square root X plus two over square root three X minus two. Minus square root two X minus one As X goes to bomb before doing any reformulation, you should always check the limit. May be it is an easy calculation. X goes to one, so we just have to substitute one into each term. The first term of the nominator goes to square three as X squared and acts are both equal to one. The second term also goes to square a tree as one plus two equals two tree. The full nominator is going to zero as the result. It might be all right if the denominator doesn't go to zero. The first term off the denominator goes to bomb three months, two weeks, 1st 1 and it's square root is also similarly, the second term also goes to bomb. This is where we can see the problem. We have a limit. A 0/0. There is no simple solution. We have to reformulate the fraction in cases off fractions off Pelino meals, we could just divide by X minus one as oneness zero feet. Now it would be really hard to find the product in which X minus one appears as a factor. For month, we must get rid off the square root, either in the nominator or in the denominator. The usual area identity can be used denominator or the denominator can take the road off a minus. Bi onda. We must more deploy I with a plus B that's try to re firmly the nominator. To do that, we multiply and divide by the corresponding A plus B. So we take the nominator, change the South direction to addition, and then we multiply and divide at the same time. We use the multiplication to get rid off the square roots. As we multiply the denominators off the two fractions, we multiply a minus bi and a plus B, so the result is a squared minus B squared. In the present case, this Miss Square root X squared plus explosive on squared minus square root, X plus two squared so the square roots drop out its verse writing brackets to avoid mistakes in the subtraction. I just did the subtraction in the next step, X squared minus one remains in the nominator. With this, we successfully reformulated the original fracture. In most of the cases, we read solution by this. Now it seems like Mystere have limit a 0/0 X squared minus one goes to zero as X goes to one. Therefore the whole nominator goes to zero. The order didn't know that the denominator of the first fraction goes to zero. So the whole denominator also goes to zero. We still have work to do. Let's repeat the trick with the denominator off the original fracture. This stand, we multiply and divide the denominator by square root three x minus to plus square root two x minus. Mom. Actually, the division off the denominator appears as a multiplication in the nominator. The main thing is to write up this multiplication and division at the same time. This time we multiply the first denominator with the denominator to drop out the square roots from the first denominator. Three X minus two minus two x minus One remains in the first denominator. After we have done the multiplication, we can do the sub direction on. We can put the second and third fractions together, the subtraction leads to X minus. Mind the denominator of the first fraction. We still have problem with the limits. X squared minus one and X minus one are both standing to zero as X goes to bomb. This makes the whole expression go to 0/0. Which means a huge question mark. However, we have already got far. There is no trouble with the square root. The second fraction simply goes to one plus one over square three plus square. We only have to deal with the first fraction, which is a fraction off polynomial. We have a drink. If death rates a 0/0 as X squared minus found it 1st 20 at 11 is a zero beat of the fully no meal. So it is going to appear at the factored format X minus one. Master Pierre. If you bribed the bonoma up as a factor for month to be more precise, X squared minus phonic was two x minus one times X plus. Mom, we can't simply pie by X minus. Mom, it's OK to do it as X doesn t question one. It is just standing to bomb. Now we can finally calculate the limit. X goes to one so we can substitute one everywhere the limit off to over square three. It's the result. It's important to see that the combination off these two matters is going to give you a result. In case of problems, including square roots and point no meals, just use them under your gauge solution. 11. Classification of discontinuities: often re calculate the limit off a function to investigate the discontinuity or discontinuities off the function. That is why it is important to know the classifications off. This continuity is this. Continuity means that in a given point off the function, the function is not continuous. It can happen if that point is not part of the domain, said there isn't any function value at all, or it can happen if the value of the function simply jumps in that point, which means that there is a sudden change in the function value. Basically, there are two main kinds of this continuity. There are these continuities off the first God. In their case, the one sided limits off the function exists at this continuity and their values are finite . So if you calculate the limit off the function from below and from above and you get a finite limit in both cases you have found. But this continuity off the first kind. However, there are two sub cases that are essentially different. There is a removable. This continuity if the left hand limit on the right hand limit is the same. In this case, the function is basically continues but there is a point where the function doesn't have a value. For some reason, this is a discontinuity. But if you redefine the function by saying that its value should equal to the one sided limits at this point the function would become continues. That is why, because it's this continuity removable. The other sub case is the jump, this continuity in case of a jump. The left hand and right hand limits exist, but their values are not equal to each other. There is a jump in the function about you. So once again, in case of a remove about this country limited, the right hand limit equals to the left hand them it. So by defining the function properly, we can remove the discontinuity at this point by not changing the function values off any other point in case of a jump discontinuity. The right hand limit doesn't equal to the left hand limit, so we cannot make the function continues by just defining its value properly. In that point, dysfunction is going to have a discontinuity independently from the function value to this continuity. Besides the discontinuities off the first kind, there's another group, the discontinuities off the second kind from the previous definition. It is obvious that we can call a second kind this continuity everything, which is not the first. Candace Continuity is a second God. This continuity. So either the right hand limit or left hand limit does not exist. Or maybe neither of them exists. This also contest the case, but the limits equal to infinity in the mathematical sense. We don't consider infinity to be a valid limit either if the limit doesn't exist at all or if it is infinite, we consider the function divergent. So mathematical e there is no limit. You can call the discontinuities of the second kind as known. Remove about this continuities or, in other words, asan. Show this continuities that miss summarize the process. First you need to find a discontinuities off a function. Then you should calculate its one sided limits. And then you can categorize that these continuities based on the one sided limits 12. Investigation of discontinuities – 1. Example: let me show you how to investigate the discontinuities off a function as this is the main application off the limits of functions. The function to investigate is X squared minus two X minus eight over X squared minus five X plus four. We should determine the points of this continuity the locations which cannot be included in the domain. Then we should determine the types of the discontinuities. Let's start the solution by looking for the points of this continuities. The nominator and the denominator also contains continuous functions for which the domain could be the whole set off rheal numbers. Travel can only occur when the denominator is zero. As in those points, the fraction cannot be calculated. So that's look for the roots of the denominator. Now we have a second order function in the denominator, so the solution can be calculated by the usual formula. For larger order equations, you should find at least a few solutions and then you can factor the polynomial to decrease the order of now, X one equals 24 x two week 1st 1 At the two points of discontinuity, these shabby investigated further. Let's check the type off this this continuity at X one To decide about the type, we must calculate the limit at X Y as X one equals two for for the limit must be calculated around four. So X approaches for Let's just try to see the limit off the function. Maybe the solution is straightforward. In the nominator, X squared goes to 16 miles to eggs goes to minus eight, so there is 16 minus eight minus eight, equal to zero in the nominator similarly began that the denominator goes zero to AC Squad goes to 6 10 minus five X ghosted minus 20. So 16 minus 20 plus four is the nominee denominator, which is really zero. So we have some difficulties, as the result is in zero where zero form, we should reformulate the fraction we can use the usual matter. So let's factor the the nominator on the denominator X minus four is definitely going to appear in their factor for the nominator equals two X minus four times X plus two on the denominator equals two X minus four times X minus. Mom, either you can calculate the roots and fried the factoring with multipliers in X minus truth for months or you should guess. But the factoring is in the denominator. We know that four and one our roots, obviously the X minus root formal gives us X minus four and X minus Fondness factors in the nominator We already knew that four is the route, so there must be X minus for you can calculate the other route or just follow my logic, but him with it I X wheat to have X squared. It is acts so you can write X in the second. Bracket them. What do you multiply? Minus Forbade to have minus eight. It's plus two. You can't write this down too, if you check minus two. X is also coming from this product as there is minus four times X plus two times X, which results in minus two X. It is important to be able to factor the Polina Mia, so make sure you have this skill. After you've got this factor form, you can simplify by X minus four only X plus two over X minus one remains, which is pretty simple to investigate as X ghost for denominator goes to six and the denominator goes to trade altogether, faction goes to two as X approaches for this result is independent, off the direction from which we approach for the right and limit on the left hand limit are both do so. There exists a finite limit for both one sided limits. This means discontinuity off the first canned at X, Big first before Furthermore, we know that the limit is the same for both directions that for this is a removable, this continuity if we defend the function as this fraction. If X doesn't for four and we defined the function value at exit first before to be plus two , then the function would be continues. Let's check the other discontinuity to it is an ax equals to one, so X approaches funding this case. Now I work a bit more. According to the official playbook, we should calculate each one sided limits separately. Now I start with limit from above. If the two limits are going to be the same than it's all right to calculate just volume it . But otherwise you should calculate to separate limits as this is going to be a red right hand limit or a limit from a bow. The value off acts is slightly greater than one, but it is approaching. Let's see the limit of the function. The nominator becomes one minus two minus eight, so altogether minus the denominator goes to zero. However, it is very important to get a proper son off the denominator as that water is the sign off the whole result. Now we are working with negative numbers as X squared minus five X plus four is negative. If X is slightly greater dumb bomb, it is easy to check. We know that one and four other two points, but the denominator changes sign. So let's take a number between violent for, for example, that state to and substituted to the denominator. Two squared minus five times two plus four Request a minus two, which is negative if it is negative, or the results between X equals to one and four a negative as a finite number is divided by a number going to zero. The result is infinite, as minus nine is divided with a negative number. The result is plus infinitive. In this case, let's see the other case, the left hand limit, which is the limit from below. The logic is the same. The only difference is the son of the denominator. If X is slightly smaller damn palm than the denominator becomes positive, you can easily check this too. For example, we can substitute X the 1st 0 The denominator becomes plus four, which is really positive. So the sign of the denominator is positive If Max is smaller Damn bomb as I sat the denominator is a positive number which goes to zero. So it's Rasa broker goes to plus infinity As there is a minor son in the nominator, the reason becomes negative. So this result is minus infinity at the end. So technically the limits do not exist as their value is infinite as the one sided limits are infinite so the limits do not exist Mathematical In this this continuity can't be off the first guide. It is a discontinuity off the second can't to be honest, we could have stopped after the first is off. If any of the limits are infinite, then that this continuity is off the second kind for sure 13. Investigation of discontinuities – 2. Example: Let's see another problem related to discontinuities. The function to investigate is X cubed minus two X squared minus X plus two over X cubed minus two X crab. We should find the point of this continuity, and then we should also determine the types of the discontinuities that's start by finding the points as both the nominator and the denominator continuous functions. The only discontinuities are where the denominator off the fraction is. Zero. So that's check that X cubed minus two X squared equals zero. Fortunately, you can immediately see that zero is a solution, so acts and even X squared can be a factor. So, in fact, art form X Q buys two x Craddick First the X squared times X minus two You can easily see from this that, except 1st 0 and X equals two other two solutions. Actually, the third solution is also acts interested. Zero. So there are two points off this continuity zero to delimit shall be determined at these points. Let's start at XY questi zero to be more precise. That's the orbit Devaughn sided limits separately. Now I start with the limit from above. First of all, it is vert ID to reformulate the function, especially that we already factor the denominator. This stamp just haps you to have a better viewpoint at the function. But now let's check the limit in this format in the nominator every term, including X, goes to zero. So the nominator simply goes to two in the denominator X squared the ghost zero. So the product goes to zero to we still need to find a sign off the product, so we should check the direction in which the function approaches. Zero X squared is obviously positive as it is ascribed term. So X crowd goes to zero as a positive number X minus two goes to minors to around which their negative numbers all together, the denominator is negative, so we divide plus two with a negative number that goes to zero. As there is a division with the number going zero, the result becomes infinite. According to the sun, it is going to be minus infinity. The method is the same with the left and limit to In that case, Ext approaches zero from below, so X is actually negative in the nominator. The terms, including X, are going to zero so the limit of the nominator is due in the denominator. X clad steel remains positive on the other factor goes to minus two. So we get the same result on both sides of zero. However, the limit is minus infinity. So this can't be a removable. This continuity as technically the tu minus infinities are not the same. Still, you can just ride that the limit is minus infinity as X approaches zero. You don't really have to indicate the van said it limits as the result has been the same for them. Now that we know the limits, we can determine the type off the discontinuity as the one sided limits are infinite, so they do not actually exist. Mathematically, this cannot be a discontinuity. Off the first kind, there can only being a second kind discontinuity. Let me remark that even if the limit is minus infinity at the two sides of zero, the two sides are not meeting each other at minus infinity. You cannot make this function continuous at X equals zero. Let's continue with the other dis continuity it is at actually crossed it to once again. We start with the limit from above. You can immediately substitute to into the nominator on the denominator. After the substitution, you can see that the nominator goes zero to cubed equals to eight minus two times two squared equals minus eight. So far it is zero. Then we subtract two on two said there isn't zero. The denominator also goes to zero, so there is an expression in 0/0 form. We should try to use the usual mattered and factored nominated and denominator the factor form off the denominators or no. So we now just need to reformulate the nominator. I have just calculated that two is the root of the nominator, so X minus two is going to appear in the factor. For I'm looking for the nominator as the product off X minus two and something else. It is not so easy to find the roots off a cubic creation, so let's be tricky. You should think about the factor, which you can use to multiply X minus two it and get X cubed minus two X squared minus X plus two. First, it is enough to get the term with the greatest exponent. Extra be multiplied with something to get X cubed. this must be X klag. What if we multiply by X clad X Times X squared gives you x cubed minus two times X crowd gives u minus two x crab. So the 1st 2 terms are already obtained. You just need to get minus X plus two extra be multiplied with something and the result is minus X, so you multiply with minus small minus two times minus one. Also gives you the plus two. So you have got everything. Now we can consent right on the limit. We can simply fight by X minus two. So only X squared minus one over X Crabtree mates. Its limit can easily be calculated. X squad goes to four so the nominator goes to train on. The denominator goes to four, so the limited function is 3/4. As exclusive to from above, However, it doesn't really matter that X close to two from above or from below. The sun is not going to change, so the result is the same. If ax approaches to from below, you could calculate in the same way. So the final form off the function also becomes X crowd minus one over X quad so it is enough to check its limit. What, together? I can say that the limit at X equals to do is going to be 3/4 independently off the direction along which the function approaches to so I can write a normal two sided limit instead of them onside limits. More importantly, there exists a finite limit, so there is a discontinuity off the first kind at X exhausted. Furthermore, it is every move about this continuity as the limit is the same if acts approaches from love from above. So if we define function value to be 3/4 at AXA, quest to the function becomes continues. 14. Investigation of discontinuities – 3. Example: I show you a special function which has these continuities dysfunction is the sine function we must didn't remember. Does sine x squared minus X minus 20 Have it's this continuities on. They also have to decide about the types off the discontinuities. First, let me talk about the sine function. The san function extract the sign off rial number there for the value of the sun X only depends on the sign off X If X is greater than zero san acts exhaustive one if excess more than zero san X equals the minus one. In between these options sine x zero If ax is zero, this is a strange function but it does have a practical meaning. Actually, the sine function is the derivative of the absolute value function. Also, some applications require dysfunction for a precise definition. For example, in case of friction or drag, we need sine function in the precise definition. Let's get back to our task and find the points of discontinuity. Is X squared minus X minus 20 is continues on the whole domain off rheal numbers, so it does not have any discontinue. It is by itself. Therefore we only have to look for the discontinuities off the sine function. Those are going to be the points. But the sign off the pulling a meal changes. If the polynomial is positive, negative or zero, there is no jump in the function value. The result of san function is constant. But for example, if the Polina meal is positive and then become serial, the san function has a jump from 10 So they're going to find these continuities. Wherever the polynomial becomes zero, we must check where the second order putting a meal equals 20 We can use the balance solution formally to find the solutions. X Van Dijk 1st 5 and X two equals two minus four. These are the solutions for the second order recreation. So these are going to be the points of discontinuity. That's the nature of the san faction. First, let's check the type of discontinuity at X equals 25 To do this, we must calculate the van cited limits. We can start with the limit from above if X because five the polynomial equals 20 If X is slightly greater than five, the Polina Mere becomes positive. So the sun function gives month As a result, you can simply substitute six. For example, it is greater than five. So it should give the right sign for the numbers above five at the sun can only change wherever the polynomial zero six square minus six months 20 equals 2 10 which is in fact positive. We can do the same for the limit from below. We need the son of the Polina meal. If X is sliced this morning down five as minus four and five other 20 bits. You can substitute anything between them to find this sign that we need, for example, that substitute exit question zero. In this case, the polynomial becomes minus 20 riches Negative. So the sine function gives minus one is the result. Now you can clearly see the bonsai did limits are different, but at least the one sided limits exist. This means that there is a discontinuity of the first kind at X in quest of five. As the van sided limits are not the same. This is a not removable this Contin 80 This is a jump, this continuity off course. This has been expected you to the definition of the san function. But we could get a result where the discontinuities removable. We expect the similar behavior for the other discontinuity. But let's breathe to the matter to make sure once again, we can check the limit from above. X is slightly greater than minus four. By substitution, we can see that the terming the brackets is negative in this case. Actually, we have already checked the sun in the interval between minus foreign plus five. So the limit from a boat is minus farm as the plane Amir listening now comes the limits from below fax is smaller than minus four. We can substitute something and find that the sign off the polynomial is positive. The sine function gives bomb. The one sided limits exist in this case two Therefore, that is this continuity off the first can't There is a jump as the one sided limits are different. This this continuity can be removed 15. Trigonometric functions – 1. Example: let's start dealing with the limits of functions that include Trigana metric functions. In the first example, we must determine the limit off sine X squared plus six x clad over X clad as X goes to zero. The interesting part of this problem is to deal with the side function. There is a special result that you can use at all times. I've already mentioned it. Among the special limits. The limit off sine X over X is one as X Ghost zero. In most of the cases, the limit off a trigger automatic function can be calculated based on this value on special limit. If a polynomial is part of the function than especially, you should think about this limit. Please know that this sine X over X can be generalized. Anything can be in the place off X as long as it appears in the argument off sign. And importantly, we should seek the limit around the point. But the nominator and the denominator are both zero. Now let's reform later function to make something like sine X over X appear by doing the divisions independently. Sine X squared over X crowd appears in the desert form the remaining part can easily be handled as six X crowd over X squared campus. Simplified, his ex doesn't equal to zero just standing to it. The second term six, which certainly danced. Six. Don't we just need to deal with the first time? You can simply use the analogy at zero Both the nominator on denominator is zero X squared takes the roll off X. So the limit off sine X brad over X squared is the same as the limit off sign axe over X as X goes zero, the first term goes to one. So the whole function goes to seven as X goes to zero. We've got the final result always trying to use the limit off sine X over X. In most of the cases, it is going to work out just fine. 16. Trigonometric functions – 2. Example: Let's see a general problem, which only includes trigger automatic functions. The task is to determine the limit off sign for X cubed over tangent five x cubed as X goes to zero. This is an interesting task. There is no easy solution was the nominator and the denominator equals zero as X goes to zero. Also, the arguments are different, so we can't really solve the problem by using trigger automatic identities. We cannot match for X cubed with five x cubed. If the dig a pneumatic identities cannot help or only hope is to use the Valium limit, which includes Sign the limit off sine X over X is one as X goes to zero. This is the only help that we have in the present case. Without any better idea, we should try to reformulate the fraction to make this special limit appear. If there is no pulling no meal in the faction, let's put some in it to get sign acts over X in the function, we should divide sign for X cubed by four x cubed. The limit of this part of the function is satin Yvonne At zero off course. We cannot just divide by four x cubed as a compensation, we must also multiplied by foot execute. This is one part of reformulation. We also have to do something with the tangent as four x cubed dance 20 at zero and tangent also tends to zero. We still have zero where. Zero. That's why I ordered the substituted tangent with a trigger automatic identity. Tangent X equals two sine X over co sign X. It is better to have signed and co sign than a tangent. We can somehow combine Sign with a polynomial, which is batter than having attention. The co sign is also better because it's limit is one. As X goes to zero, they just go sign eggs. But consign five X cubed also goes to one at zero. So that's all right or we have to do is to determine the limit, which includes for execute and science five X cute. Let's repeat the previous streak. This time I multiply and divide by five x cubed. The multiplication part can put together with sign so five x cubed over sign five x cubed appears the limit off. This is going to be from the division by five x cubed can be done on the other polynomial as we can simplify by X cubed. But the previously formulations Very rich. The product that consists off four terms that will have easily rated limits. We can just call like the limits and calculate their product. The first term has been made to be like sine X over acts for X cute takes the roll off X But anyways, the limit is the same. The first arm goes to one, eggs goes to zero. It is also very important that X goes to zero and boost the sun and the appointment zero. At that point, don't forget to check this part. The second term goes to 4 50 as it is just a constant number. The ter term is similar to the 1st 1 but this time the polynomial is divided by sine and not the other way around. There is no problem with that. We know that Sign five X cubed over five x cute goes to one as X goes to zero Here we would like to know the limit of the reciprocal off that easily The limit off five X queued Hoover sine phi X cubed is arrested broke off the limit off sign five X cubed over five excuse. So it is going to be run over von Witches. Generally, it doesn't matter if we divided pulling a meal or with sign If there's sine X over X, so there is ExcelAire sine x zero. The limit is always going to be fun. The fourth term is easy. Five. X cubed goes to zero as X Ghost zero s co sign zero is one. We know that limit as X goes to zero close on five x cubed goes to one. The result is 4/5 that is the product of the 14 here. We also use the fact that the limit of product is the product of the limits off the terms in the product. 17. Trigonometric functions – 3. Example: let me show you a problem, which is even more twisted than the previous ones. The limit off X minus two over sine X minus two times arc. Don jump X squared over two minus X squad has to be calculated as X goes to to you can see a product which should really be investigated term by term. It would be very hard to come and sign and act on John by the help of Trigger Matic Island is especially in this case, but arguments are different. Let's start by calculating the limit off the first time. This is similar to the special limit off San X over X zero. First of for the rest, it broke all of the battle clematis taken. That's all right. The limit would still be bomb. Secondly, the limit is not taking a zero, but to the most important thing is that boost. The nominator and the denominator must be zero wherever the limit is calculated. Now they are zero at two, so we can use the vandal result. To see this better, you can introduce the automotive arrival. Let's say that my equals two X miles by substituting righted limit the recipe broke off the value in them. It appears my over signed by is investigated as why goes to zero. It is important to revive. X goes to two as by ghost zero. Don't forget that if you write this up correctly, you can really see the limit. We can use some worry formulations to give very precise and nice answer. If we divide nominator denominator by why I signed by over by appears, we know that it's limit is one us via go zero. This is what we can actually state as the rest of local off one is was one. The limit off. The original expression is too. Now we have got the limit for the first term off the original product. That's concentrate on the second term that includes Arc Tangent. Let's try to work as clearly and easily as we can, even the argument of Arc Tinge and this quiet complex. So let's just calculate its limit first on down to get back to our engines. So the limit off X crowd over to Manus X crowd is served to you can clearly see that X equals two is a place of this continuity for this function the denominator is zero at X, Exhausted do. Therefore, it is more precise to calculate the van sided limits. Off course. You can try to calculate just volume it, but be careful about the science. If you do so now, we calculate the limit from above. Denominator goes to to scratch, which is for the denominator goes to zero, and it is definitely positive, as two minus X is squad. All together for a finite number is divided by a number that is slightly greater than zero but goes to zero. The limit becomes plus infinity, as we divide with a positive number under nominator is also positive. If the square wasn't in the denominator, we would have to be very careful with the science. In that case, the limit from above would be minus infinity as two minus X went negative. In that case, that too one sided limits would not be well. Now let's check the limit from below. We suspect that the limit is the same as using square terms emanates the effort off signs. Everything is positive. The result is really the same. The nominator goes to four. On the denominator is a positive number that goes to zero The result of division by such number results. In a great number that goes to plus infinity what together we can say that the limit is infinity. Independence laid off the direction in Vitry approach to the one sided limits were the same . This is not necessarily true, so be careful with such problems. Now this is a good news for us as this limit shall be substituted into Arc Tangent so that see the limit off arc Tangent X credit where two minus x class. We now know that the argument goes to plus infinity We just need the value off arc tangent at Plus Infinity. If you know the Arc Tangent function, it is straightforward. I didn't suggest knowing the main properties of trigger automatic all functions, including the shape of the graphs Arc Tangent goes to buy over to as the argument goes to infinity. Actually, the Arc Tangent function can be booked in bar mirroring the dungeon function to the Y equals two X line. You can get the shape of the function from this fact or so if you know the tangent that goes to infinity at BioWare too, and you know the embers of tangent is Arc Tangent. You know that Arc Tangent must be by over two at infinity. Anyways, we've got the result the limit off. Both terms are known in the original product. The first start goes one on the second term goes to buy over to that, for the product goes to one time spyware to which is by word. 18. Limits based on Euler's number (e) – 1. Example: In some special cases, the limit off a function can be traced back to the definition off Eilers number. I have shown you the corresponding special limit earlier, So now let's see how and when we can use it. The limit off X plus five over X plus. Want to the power of two X must be calculated as X goes to infinity. The function is a polynomial fraction race to a power which also contains the variable. If the function is something like this and X goes to infinity, we can apply the value limit one plus one over X to the power of X goes to E as X approaches infinity. It is very important that we can only use this limit if X goes to infinity. This constraints the application to a very narrow set off problems steal. This is a matter advert to mention Let's try to re firm later Original functions The nominator and the denominator are in the same order. However, we need a constant plus. The fraction by the denominator is a tire order Done denominator. We must use polynomial division If we divide X plus five by X plus Mom, we get one as a result. And the remainder is for because knowing the polynomial division, you can just reformulate the nominator by including the denominator. X plus five equals two x plus one plus four X plus one over X plus one gives you one and for over explosive on remains. This must be the first step at all times. Then we must ensure that the constant is one. It is all right now. Then we had one over something. Now we have four over something. Let's divide the nominator and denominator by four. Don't have one over something as the second. Now the one plus 1/1 quarter times X plus one is raised to the power of to axe. There is one step left in the Vernon limit. The term used for division on determine the power were on the same. Let's riff right to acts by including the current denominator In it. Two X equals to eight times one for 12 times X plus one minus two. First of all, we need the denominator in the power term, we look for the proper more triplicate er to match the number off, axes down video with the constants by adding or subtracting them. The necessary expression appeared in the power term. But it's better if we do some worry formulations Just go according to the poverty identities. The multiplication in the power term can be substituted by taking the power off the whole term. Then, for the eight, power is taken. The minus two means that we need to divide by the base of the exponent vice. If there isn't at the show, you should multiply most importantly week after the end. Now we can calculate the limit. The Valent limit appeared so determined. The scratch brackets goes to E 1/4 times. X plus one takes the roll off axe. This part is raised to the eighth power for so the limit is also raised. The age barber. That's all right. The second part is quite easy. The denominator goes to bomb one over acts would go to zero. As X goes to infinity, so does the fraction. Inside the denominator. We had this to one which means one. By taking the second power off it, Mystere get warm. Therefore, the final result is eat the father of eight once more. The second term is one because there is a finite power off the expression that goes toe bomb. In these cases, the limit is always from in the first term. The limit off the expression inside the rounded brackets is also bomb. However, the power off that is infinite. That's why that whole term can be different from bomb. As we know the limit of that becomes E. Please don't get mixed up with these. The limit off one plus one over X. The power of X is always e. If the power is finite like 123 the limit is fun as you would expect. 19. Limits based on Euler's number (e) – 2. Example: Let's see another example that oil loves number appears in the solution. The limit off three X minus five over X minus one to the power of acts must be calculated. Let's use the same matter that's before the Vanna Limit for this type of functions. Reads as farm plus one of Rex. The bother of X goes to E, but X goes to infinity. Let's get or function in a similar shape. The denominator can be included in the illuminator by looking after ex three times. X minus one provides three acts. The remaining constants can be found now we should subtract to get the original dominator. Now you can easily see the three times X minus one over X minus one EQ boss too three and there remains minus two over X minus one. We would like to see one plus one over something in the brackets. To get that, we must get three out of the picture become bring it out of the brackets. Actually, by taking three out, we get three to the power of X, as everything was in the past. Power inside the brackets, a stretch of the power of X is uses a multiplier. We should divide the original term by it to get a quality every term in said, the brackets is divided by tree. That's how we get one in front. That's importantly, the second term becomes minus 2/3 times X minds. Now that one is in front. We would like to see one plus one over something now minus two is in the place of one. So let's divide denominator and the denominator by minus to the denominator becomes minus 3/2 times X minus one. Now everything is well inside the brackets. We just need to make sure to have the proper power term. We need to revive right x in a suit above A to make the denominator appear in the power bi certainly Freida Denominator into the power to get one times X back, we need to multiply the denominator by minus 2/3, the coefficient of X becomes bomb. Then we deal with the constant. There is zero constant in the original power. After the multiplication, there is minus one in the new power. So we need to advance to compensate. Now the denominator is in the power and the whole expression it first the original off task . Formally we can re fried the terms. The bomb plus one over X to the power of ex form appears in the scribe brackets. It's popover is minus 2/3 on There is a need for an extra multiplier. So we multiply pants. Now we have the final shape of the function. We just have to calculate the limit of each term. The one in the square brackets goes to e. It was the whole point of everything to get this. The third term just goes to one as it is one plus something that goes to zero at infinity Finally, three to the power of X goes to infinity as X goes to infinity all together we have good infinity times e to the power of manners to over three times Mom, the result is infinity. Actually we could have noticed this earlier they could look at the original function. The expression inside the bracket goes to three as X goes to infinity. This is raised to a power that goes to infinity. This indicates that the results must be infinity 20. Limits based on Euler's number (e) – 3. Example: Let's see one more example of fusing the Eilers number two silver problem. This time the limit off X squared plus three over X squared to the power of X squared shall be calculated as X goes to infinity. The usual ban on limit shabby used. Therefore, we should reform late to have the bomb plants one over something to the power of the same something this part off the expression is going to have to limit e. We can divide exc Reddin tree separately by ex crab. This means that we have one plastered over at scratch. Within the brackets, you can see that the term in the brackets goes toe one as X goes to infinity. If the limit inside the brackets is more tambon than the whole limit would be infinitely as infinite, barbarous, taken. If the expression inside the brackets is smaller, down form on the infinite pavers taken than the limit would be zero. We have seen the first case in the previous example, but now let's just don't in you by turning the nominator into bomb, let's divide denominator and denominator of the fraction by tree. Now there's one plus one over something the next goal is to have the same denominator in the power we've right 1/3 times X squared into the power. This should be multiplied by three to get X crowd back and insure the quality. Basically, we have the result. The multiplication battery in the poverty could be considered as taking the cube off one plus 1/1 3rd times X squared to the power off Want er times x clad therefore the limit Is he cute? Let me add just mom Mortar This mattered always works when there is one plus one over something to the power of the same something. And this something that is in the denominator and the power goes to infinity. So if there is a polynomial in the denominator on the power, you are good to go. However, let's say there is one plus one over sine X to the power of sine X. In this case, Sine X does not go to infinity. Therefore, the Valium limit couldn't be used. Be careful about this. Only used This mattered if the poverty term on the denominator is polynomial 21. L’Hospital rule: the low Peter Rule is a magical tool in case of problematic limits. Differentiation is needed to reply the Lopata rule. Differentiation is a huge topic that is based on the limits of functions as differentiation is a higher level knowledge down limit calculation. The locator rule is way too powerful for the previous topics that I've shown you. If you don't know differentiation yet, don't worry. The things you have learned so far is enough for you. I show you this tool and its application to make the course even more complete. I provide the basic derivatives and basic tools to you. So without further understanding off differentiation, you can use them in this way. You are going to understand this section, but really only use low Peter rule in exams. If you are able to use it, it is really an ultimate teachable. Let's get to the low Peterle. The Lopata rule states that, in case of problematic fractions were either denominator and denominator. Both stands to zero or both dance to plus or minus infinity. The limit off the fraction equals to the limit off the fraction that is made by dividing the derivative off the original nominator by the derivative off the original denominator. So after coma, X means the derivative off FX and G common X means the derivative of G X once more, there is a problem where you should calculate the limit of fraction. You find out that the nominator F X and the denominator g x our pope is going to zero or both are infinite as eggs goes to appoint indicated with a you just simply don't know what's the limit off 00 or the limit off infinity over infinity. We have seen such problems so far. We have used some tricks on reformulation. Now the locator rule lows you to calculate to derivatives, which is a very easy step. And then we can calculate the limit off a new fracture. If this fraction is problematic, too, then you can make one more step. You can go on and on until you get the result. This matter is very advantageous. First of all, you don't have to think it over. If you have a problem, you use derivatives and the problem is solved. Let me say a few words about applications obviously views the locator able to calculate the limit of problematic functions, either. If this is a 0/0 type fraction, or if it's an infinity over infinitely type fraction, you can use the Lopata rule in case off such functions. However, other kinds of problematic limits can also be calculated by the half. Off the low petero This is done by reformulation. I'm going to show you examples on tricks in the upcoming lectures. Once again, if you are a university student, only used this matter if you are a load to use it, your teachers wouldn't be happy to see such a solution if they wanted. Desk your skills reformulation on basically mitt Calculation. On the other hand, if you just have to deal with your life problems, this is the rule that you want to use. 22. L’Hospital rule – 1. Example: Let's see a basic example of the application off the Lopa. Terrible. The limit off course. An X minus one over X crowd shall be calculated as eggs. Ghost zero as X goes to zero. Co sign eggs goes to farm in the nominator of ah minus one goes to zero in the denominator X crowd goes to zero. Therefore, this is a 0/0 type limit. In such cases, the Lopata rule can be applied. This limit equals to the limit off the nominators derivative over the denominators derivative. So we must differentiate the nominator and the denominator separately. And then we should investigate that court. Let's start by debating the nominator. The derivative off course Sine X is mine of signed acts. According to the list off basic derivatives, the diverted off one is hero as the derivative off a constant number is always zero. So the nominator becomes mine of signed acts. The denominator is X scratch, which is a polynomial. The derivative off a polynomial leads us the following. We ride down the original proper as a multiplier and then we decrease the power bi from the derivative off X glad becomes two times X to the power off bomb. So it is two X according to the Lopata rule, the limit off minus sign acts over two X equals to the limit. Of course. An X minus one over X crab. Let's check the limit. Now Sign X Ghost zero s ex ghost zero and also to wax goes to zero. So once again, this is a problematic limit. Off course there's about limit that we know off the dim it off. Sign axe over axes one at zero. Now, let's not use that. Another practice they use off the GOP. Terrible. We have a zero where? Zero type limit. So we can reformulate once again by taking the derivatives off the nominator and the denominator. The derivative off sine X is consigned X. Therefore, the new nominator is minus co sign axe The day votive off X is bomb or actually one times after the power of zero, which means from the multiplier to just remains a multiplier. So the denominator is two times one, which is two. Now we can check the limit co sign X goes to one as X goes to zero. Thus the nominator goes to minus one which makes the coach and go to minus one work too. This is the limit off? No, just last fraction. But also it is the limit off all three crows and 23. L’Hospital rule – 2. Example: Let's see another example that the low Peter were can be used in this example. The limit off two X minus four Squared times E to the power of minus tree extra. Be determined as X goes to infinity. First of all, let's check the limit by investigating the terms in the perfect. The first term is obviously positive, as two X minus four describe within the brackets, two x goes to infinity, so two X minus four also goes to infinity. The second term goes to zero as the proper term goes to minus infinity. It's easy to see that ministry acts goes to minus infinity on. We know that the actual essential function goes to zero as its exponent goes to minus infinity. So there is an infinity Times zero type limit. For this, we don't have any matter. The Lopa terrible can only be applied for fractures, so let's make a fraction out of the product. If we take the recipe, broker off one term and divide with it, we get a fracture. It's much more easy to take the reciprocal off the explanation. Shutter on one hand, one over. Eat the power off minus reacts simply equals two into the power of three acts, on the other hand, the day. But if becomes much easier in this way, sometimes it's hard to decide how to make the fracture. You can just try both ways. You can take the recipe, broker off any of the terms and divided it. However, if the solutions had, just try switching to the other possible a fraction and investigate that now we divide the polynomial with reciprocal off E to the power off ministry acts. So actually, we should determine the limits off two X minus four squared over E to the power of three ax . This is the same as the original function. Distraction is on infinity over infinity type function was the body. No meal and exponential terms goes to infinity as X goes to infinity. Texted is we can apply the Lopata rule. That's terrified. The nominator can either be dead elevated by applying the change will or by expanding the expression by formally calculating the scratch off two X minus. For now, let's choose the change. In case of the change Will we must first debate outer function on then the inner function. The outer function is taking the scratch off the term the derivative off X squared is to axe. Similarly, the derivative of something scratches two times to something. That's why we have two times two x minus, for this comes from the outer function. Then comes the inner function. Now it is the Vung inside the brackets, which is two X minus four. Let's differentiate that the derivative off to acts is to on the derivative off minus 40 That's why we get the two as an additional multiplier. We work similarly in the denominator here. The outer function is the exponential function and there is an inner function in the power of the explosion function. A Street acts is in the power for the derivative off the exponential terms is itself so er three x gives us e r d X as an exponential terms. Then we multiplied this with the derivative off the inner function, which is three acts the derivative of three X tree. That's how we have got the new coach chant that we can investigate. The limit is still on infinity over infinity type limit in denominator, X ghost infinitely on the remaining constant student bothered that in the denominator. Exponential terms Steer goes to infinity. Bestir need reformulation. So that's applied the Lopatto Guardiola gum. We should differentiate the nominator and the denominator in the nominator. You can do the multiplication eight X minus 16 shall be differentiated. The derivative of eight acts is eight on the derivative of minus 16 0 So the new nominator is eight. In the denominator, we do the exact same us before the derivative off er three axis. Three times he had three acts. As the outer exponential function remains as it wants on the derivative off three X gives us the multiplier of the other multiplier just stays very it for us. Now we can check the limit, eight goes to eight and the denominator goes to infinity. As the axe financial term goes to infinity, even X goes to infinity so finite number is divided by an expression that goes to infinity . In such cases, the limit off the quot and is zero. As you have seen, we can use the Lopa terrible multiple times to reach a solution. If you see a Pelino meal, andan ex financial or Trigana Matic term, the Polina meal is going to disappear by taking multiple derivatives. So just go ahead and take Morty purse, perhaps to diminish the Pelino meal and get a result. 24. L’Hospital rule – 3. Example: Let's see a new example. The limit off X over X minus one minus one over natural logarithms axe. Shabby, Determined as X goes to one. Let's check the limit like this. X obviously goes toe, therefore, X minus one goes to zero. Natural algorithm goes to zero at one suppose quotient. Goto Vanover zero. If we divide the finite number by an expression going to zero, we get a quotient that goes to infinity. So we have an infinity minus infinity type limit. There is really nothing to say about it. We must reformulate as we should reformulate. Anyways, let's make one fraction out of the whole function because we can only use the low Petery Well, in that case now we have two fractions. The best way to make one out of them is to find a common denominator by multiplying the denominators. So the new denominator is X minus one times natural algorithm acts to bring to the first fraction into this fraction. We should multiply the denominator on the denominator by natural logarithms acts. That's why you see X times natural algorithm, accent, denominator, the nominator and denominator off the second fraction should be multiplied by X minus one to get the proper denominator. That's why you see X minus bomb in the common nominator on it is subtracted as the second fresher was subtracted. Let's check the limit now. Most natural logarithms acts on X minus bongos to zero as X goes to one. Therefore, we have a 0/0 type limit. As the nominator reads as zero minus zero on the denominator reads as zero times zero. Now we can use the local terrible as devotion for fears, the requirements. That's very afraid. The nominator. First we go term by term. X times natural logarithms X should be differentiated by the half. Off the product rule. First X is Derek Baited, which gives us mom. This is multiplied with natural logarithms X that we left unharmed. Then we've right eggs down and multiply it with the derivative off natural logarithms acts that derivative advanced one over X. The square bracket contains the derivative of the product. Then we differentiate minus X minus one. The derivative off axes one on the day already off 10 So we get miners on minus zero. Now we can continue with the denominator, but we also have to apply the product rule. The derivative off X minus one is bomb. This is multiplied by natural logarithms. X. Then we write X minus one down on multiply it with the derivative off natural logarithms acts that is one of Iraq's. That's how we have got the derivatives before checking the limit. Let's do the operations in the nominator x Times one over X equals to one. So there is a plus von and the minus one. So these drop out on only natural logarithms. X remains in the denominator. Natural robot litem X plus one miners, one over X remains. Now we should try to them. It's natural logarithms X goes to zero at bond, X goes to bomb, the nominator goes to zero, and the denominator goes to zero plus one minus one over vom, which is also zero. This is still the 0/0 type limit this year. Have work to do. We should apply the Lopatto really advance again. That's very great denominator. The derivative off natural logarithms X is one over axe. This also appears in the denominator. The denominator becomes one over X plus the derivative of bond, which is zero minus the day. Everybody off! One over. X the deputy off one over X is minus one over X squared one over X is technically actually the power of minus Mom. We should write down minus mom as a multiplier on decrease the power from minus one to minus two after the power off minus two means one over X that Now let's check the limits. Once again, X goes to one. This is what we substitute. The nominator goes one on the denominator goes toe one plus mom. Therefore there is a finite limit. The original function goes to one over to as X goes to bomb. 25. L’Hospital rule – 4. Example: in this lecture a Chilean additional useful trick that you can use in case of other applications to we should investigate. Three acts the power of sign for acts were seeking the one sided limit. Iran zero. So we only have to calculate the limit at zero from above. Both reacts and signed for eggs. Guilty zero as eggs ghosted. Zero. So we bumped into a zero to the power of zero tax limit. This is very probably get really angry at your teacher. This looks like an insane problem. Three. X just goes to zero. But as there is a positive power term that goes to zero, the whole expression could go toe bomb. The limit could either be zero or bomb, but it's hard to decide by taking a look at it. If we seek the limit, even from below the poverty term, could be negatives to the reciprocal off Zero to the power of zero should be taken, which could even lead to the limit off infinity. I just told you this to see that the behavior can really differ on the two sides of zero. That's why they're only calculating advance sided limit. But what on earth Can we do about this? According to that, a big we are going to use the low Puteri with somehow. But how? The main problem is that there is a varietal at the base of the exponential terms and also in the exponent. If you ever see a problem like this, you can use one great trick. Take the natural logarithms off the whole function and then raise the natural logarithms to the exponent off E. As the exponential function and the logarithms functions are inverse functions, this is just a former change. The result of the operations has three acts. The power of sand for X. However, now you can use the loo. Great make identities. If there's something raised to a proper inside the algorithm, you can bring the exponent down and use it as a multiplier of the logarithms. So natural rogel it on t X to the power for off sign for X equals to sign four X times natural logarithms reax. This is a product that you can handle fairly easily to find a limit. We first find the limit off the exponent and then we can check the limit off the axman and shorter. So that's just investigate. Exponent. What is the limit off sand for X times Natural logarithms three acts as acts Ghost zero from above. Signed for X goes to zero and that you are logarithms three x goes to minus infinity. This is a zero times minus infinity type limit. Let's make a 0/0 type or infinity over infinity type limit and use the Lopa terrible. So that's make abortions. To get that, we either divide sand for X by one over natural logarithms three ax or we divide natural logarithms. Three acts by bond over sign forex. You can try either way. If I were you a choose deriding by one over. Signed for X. So let's check limit off Ellen three X over one over San Forex. Why do I choose this one? When you dive eight natural logarithms, it becomes one over X. So we get rid off the logarithms. If you do this in the other very around. Natural logarithms sticks in the function after derivation on makes the steps even harder. But you might also get the solution that ray. Anyway, let's now concentrate on finding the solution in this way, logarithms goes to minus infinity and sign goes to zero at zero. Therefore, the nominator goes to minus infinity and the denominator goes to infinity. This is fine. We can use the low Peter rule In cases like this, that's their abate. The derivative off natural logarithms three acts is calculated by the change will The outer function is the natural algorithm. The derivative off natural logarithms X is one over X. So similarly now we get 1/3 acts. As a result, then this is multiplied by the derivative of the inner function. Reax. The director of three access train So altogether began three over to acts which big funds to one over X. The denominator is even more complex. The change rule is applied. The most outer function is that sign for X is on the power off minus mom. That's what recipe broken means. The devotee ive off next D minus Bon is minus one times extra power of minus two. This gives us the minus sign and sign for X squared in the denominator. Then the second out there function is the sign the deliberative off sign acts isco Sanex, This gives us co signed for X now as four x is in the argument. Finally, the inner functions for acts that everybody, that is four, which appears is a multiplier. Now we can reformulate the expression a bit. 3/3 drops out. We put acts into the denominator and we bring sand for X scrapped the nominator. This is just formality. Now check the result. This is zero where Ciro type limit sand for X and X are going to zero. Therefore, both the nominator and denominator go to zero. We should do one more staff with the Lopata. I suggest making the re formulations before the derivations because it would be insane to continue with the function that we go after the last elevation. Let's make her life easier, as much become so. That's very, very denominator on the denominator of disk oceans. The nominator contains the second power off sign for acts which cometary waited by the half of the changeable Exc squad requested to act. So the second power gives the students sign for Rex than the derivative of sand. Four. X gives us co sign for X. Finally, the derivative before X gives us for in the denominator the productive rule. ISS used the derivative off minus four axes multiplied by Cosan for axe. The derivative off minus for X is four minus four. Then we add minus four x times the day everybody off co sign for axe. The derivative off Cosan gives us minus sign for acts on the derivative of the inner function gives us for as a multiplier. Watch out for the signs. The second part becomes positive as Dever to minors signs. Now let's change the limit. Once again, the nominator goes to zero assigned for exposed to zero. Because I'm four x goes to one and there are just constant multipliers On top of these, the denominator goes to minus four times one plus zero as co signed for excused one and san for X Ghost zero. So zero is divided by a finite number the limited zero. So we have got the limit off sign for X times Natural logarithms t ax it. Zero In the original problem, this only appeared as excellent off e. So let's get back to the origin off problem the limit off. Eat the power of sand for X times Natural algorithm Tree acts must be calculated now we know that explanation goes to zero as X approaches zero from above. If the exponent of E goes to see a row, the exponential term goes too far. So the final result is one that's the limit off. Three acts the Parmer off sign for axe. 26. Thank you!: Congratulations. You have reached the end of the scores. Thank you for the retention on first thing with media the and of course, and happy that you choose to learn with me. And I hope you have enjoyed the lectures. If you haven't completed the class project yet, go ahead and try to solve the problems on your own. If you find a challenging task, you can share it with others. We can solve it together. See you around.