Roberto s Notes on Dierential Calculus Chapter 1: Limits and continuity Section 1 The concept o limit What you need to know already: All basic concepts about unctions. What you can learn here: What limits are and why we study them. Imagine the ollowing eperience, one that you have probably had. You need an item o a certain kind, you go looking or it in the appropriate store and you discover that the store has all kinds o other items that are similar to what you need, but not eactly the item you want. You could settle or net best thing, but you don t eel happy about it, nor do you understand why what you need is not there. It is a reasonable thing that you are looking or and it ought to be there, so why can t you get what you need? In the world o calculus, this is the situation we ace when we realize that we need a limit. We need to compute a certain quantity through a ormula that we may know, but that ormula does not work eactly or the numerical value we need. Let me give the most typical eample. The central problem that originally motivated the development o calculus was that o computing the slope o the line tangent to a curve at a given point. Tangents had been studied or a very long time and some ingenious mathematicians had igured out ways to compute such slope in a ew special cases, but not in general. And yet the thing is there in ront o our eyes, we know it eists, so why can t we igure out how much it is? Add to this the act that, i and y represent concrete quantities rom applied settings, the slope we are ater can be interpreted as the rate at which y changes with respect to. Thereore, solving this problem could have many very useul applications in physics, chemistry, architecture, astronomy etc. So, what is so diicult about it? Let s review a ew known and relevant acts. P Deinition The slope o the line joining two points, and Q h, h P on the graph o a unction y ( ) is given by: rise h m run h Q P Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 1
So, as long as both and +h are in the domain o the unction, we can compute this slope and the equation o the line joining the points, no matter how small h is. Let me pause or two important announcements Deinition A line joining two points on the graph o a unction is called a line secant to the unction, rom a Latin word meaning cutting through. has a nice, smooth graph or most values, but has some strange, mysterious behaviour at some special value c. What is happening there? In order to describe what is happening, it is useul to adopt a common, but oten misunderstood terminology. Deinition To say that a quantity q, be it independent or dependent, approaches a certain value c, means that we can make q arbitrarily close to c. This is denoted by an arrow, as in: q c. Knot on your inger From now on, we shall use the letter h to denote an arbitrary real number that, or whatever reason, we want to be as small as needed. In usual algebra, the issue o approaching a value does not arise, since we are concerned with evaluating certain quantities and unctions eactly at certain values. However, the concept o limit arises in this contet: when we cannot get to a value eactly, can we approach it? To rephrase what I just said in more technical language, we can compute the slope and the equation o a secant line or arbitrarily small h, but not or h 0, since in that case the denominator o the slope ormula becomes 0 and the raction becomes undeined. But that is eactly the value we need or the tangent line! The store has something very close to what we want, just not what we want eactly. Once you see the solution to this problem, it will seem like a rather simple idea, yet it took thousands o years or mathematicians to ind it. Take that as a warning: though simple in the way it will be presented to you, the idea o a limit is not intuitive and neither will be many o the computational methods we ll use or it. In act, the rigorous logical oundations o the idea o a limit are very comple and are still subject to debate. But we ll not worry about it and we ll try to ocus on the intuitive meaning and the relevant computational methods related to it. Sometimes, the way to solve a non-intuitive problem is to make it more general, since the additional details may distract us rom its key eatures. So, instead o ocusing on computing slopes, let us think o a unction that can be evaluated and Deinition The problem o inding the limit o a unction y ( ) when c reers to the problem o determining whether approaches some value Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page as approaches c, regardless o whether eists or not. c Here is a slightly more ormal description o the concept, together with its standard notation.
I, as approaches c, Deinition approaches or even becomes identically equal to a inite number L, we say that such number L is the limit o as approaches c and we write this as lim ( ) L : L c c I no such number L eists, we say that the limit does not eist and write this as lim ( ) DNE. c Please notice that this is not a rigorous description o the concept, but it will do or now. You can ind a more ormal version o this deinition in any traditional tetbook or even on Wikipedia! A more important issue to consider at this point rom the practical point o view is the ollowing. The limit o a unction Warning bells can be analyzed or any value c, be it in its domain or not. However, a limit is only worth investigating when something unusual and interesting happens to the unction or values o near c. I like to reer to a limit computed at a value c or which nothing interesting happens to the unction as a boring limit. Later I will give a more ormal interpretation o what this means, but or now, keep in mind that computing limits is only worthwhile at values where something intrigues us. Eample: lim 3 Technically speaking, asking what this limit is ine, but also boring: or values o close to, the unction takes values close to 3, the closer the better. But calling on the machinery o limits that we are about to develop or this question is a waste o time: nothing interesting is happening to this unction at this particular value. Eample: 1 sin lim ; lim ; lim 0 0 0 These are worthwhile limits! In each case the unction in question is an interesting unction that is deined or all values o, ecept 0 : why? What is happening there? We shall discover later that in each case a dierent phenomenon occurs, and we shall discover that thanks to limit methods. Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 3
Eample: lim 5 This limit is boring or an even more undamental reason: the unction is not even deined or values o close to. We can certainly state that lim 5 DNE but the limit does not eist because there is nothing to compute near. That makes or a boring limit. And, just to keep the issue vivid in your mind, let me go back to the original motivating eample. Eample: limmh 0 0 c h c lim h We can think o the slope o the line secant to and a nearby point c h, c h through c, c as a unction o h, a unction that is undeined just at the value we want, that is, at h 0. Computing this limit is very important and useul. In act, it is the key idea o dierential calculus! And now a very important consideration related to an error that is commonly made by beginners, and whose avoidance is linked to a clear understanding o what limits are all about. Warning bells The process o evaluating a unction at c and the process o computing the limit o the same unction at the same value are not equal as processes. Although it is true that, or most o the unctions that we shall use and or most o the values in their c lim, this is a special domain, c property (called continuity) that cannot be taken or granted. Evaluating a unction determining the value Knot on your inger eactly at that speciic value. at c means c that the unction takes Computing the limit o at c means inding what value the unction approaches near that value, that is, as c. As you delve deeper into the study o limits, you will become better able to identiy at a glance when evaluating the unction and computing the limit lead to the same result. But or now, be careul about the dierence and try to stay out o the group o students who will make the mistake o evaluating the unction whenever asked to compute a limit. Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 4
Eample: 3sin( 1) The only value or which computing the limit o this unction is not boring is 1, since the denominator becomes 0 there. Notice that we cannot evaluate this unction at 1 does not eist. But by looking 1, so at the graph o this unction we notice that or values o close to 1, the unction seems to get closer and closer to some value, possibly 1.5. 3sin 1 It turns out that indeed lim 1.5, so 1 here the limit eists, even though the unction does not. We need a little more work on limits beore we can conirm this act, but the important point to remember or now is that evaluating the unction and computing its limits are not the same thing. Notice that the deinition o limit assumes that when is close to c, the same value L is approached, regardless o whether c or c. In some situations it is useul to isolate each o these two cases. I, as approaches c while remaining larger than it c approaches a inite number L, we, say that L is the right limit o lim ( ) L c lim c lim and we write: Each o these limits is called a one-sided limit, while a usual limit, in which we epect to approach c rom both sides, is also called a two-sided limit. c Deinition I, as approaches c while remaining smaller than it c approaches a inite number L, we, say that L is the let limit o lim ( ) L c and we write: The two-sided limit o Technical act at c eists i and only i both corresponding one-sided limits eist. In that case: lim ( ) lim ( ) lim ( ) L c c c Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 5
Obviously, because o the deinitions we are using, i the two-sided limit eists, so do the one-sided limits and all these values must equal each other. But here is an eample where the one-sided limits help us decide about the two-sided one. Eample: cos i 1 i 1 As we cross the value 1 the rule that identiies the unction changes, so this is a value or which computing the limit is worthwhile. But does that limit eist? I we look at the let limit we have: lim lim cos cos 1 1 1 The latter is a boring type o limit, as is the one we obtain or the right side: lim lim 1 1 1 Thereore both one-sided limits eist and are equal. Hence the two-sided limit eists. Here is the corresponding picture, showing that the graph is indeed approaching the value o -1. Eample: cos i 1 i 1 In this eample, just like in the previous one the value 1 is not in the domain o the unction and both one sided limits eist, but they are not the same: lim lim cos cos 1 1 1 lim lim 3 1 1 Hence the two-sided limit does not eist: which o the two limit values would we use? We shall investigate all the possibilities in the net sections. In conclusion: Technical act I both one-sided limits eist and are both equal to the same value L, then the two-sided limit also eists and equals L. I lim does not eist, then the corresponding c one-sided limits may or may not eist, but i they both eist, they cannot be equal. On the other hand Now that we have established the basic terminology and notation, we can look or methods to compute limits, so that we may look at more illuminating and interesting eamples. We shall start that in the net section. Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 6
Summary A limit addresses the question o what is happening to a unction at a value or which the unction cannot be computed or some strange behaviour occurs. The limit notation is concise, but must be interpreted properly in order to capture its meaning. Several details eist in the notation o limits and they must also be properly used and interpreted. Common errors to avoid Do not conuse evaluating a unction and computing a limit: they are dierent processes. Learning activities or Section D 1-1 Review questions: 1. Eplain what the slope between two points is and why it cannot be computed at a single point. 4. Describe the dierence between limits that are worth computing and those that I call boring.. Eplain what it means or a quantity to approach a given value. 5. Describe and eplain all notations used with limits. 3. Eplain what it means to ind the limit o a unction as its variable approaches a given value. 6. Identiy the relation between the eistence o a limit and the eistence o the corresponding one-sided limits. Memory questions: 1. What is the correct notation or the limit o a unction at a? 3. What is the correct notation or the let limit o a unction at a?. What is the correct notation or the right limit o a unction at a? 4. I lim, what can we say about lim L a a and lim a? Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 7
Computation questions: For each o the unctions provided in questions 1-36: a) Identiy any value or which the limit is worthwhile computing and eplain why it is such. b) Use proper notation to identiy any such limit. c) Determine i one-sided limits are justiied and why. You may use a graphing utility to assist you in identiying the potential values, but you will be epected to justiy all your conclusions algebraically. 1. y 3 1 1 3 8. y 3. y 5 3 4. y 5. 1 4 y 4 6. 4 3 y 7. y 1 4 8. y 3 1 4 4 9. y 4 10. 11. 1. 13. 14. 15. 16. y y y 3 3 6 3 y e 1 y y e e 1 1 1 1 y e 1 17. y e 3/ 6 9 6 18. y e 1/ ln 1 19. y 4 0. y ln 1 / 1.. 3. 4. 5. 6. y 1 y 1 1/ sin y 1 cos cos 1 y cos 1 y sin sin y Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 8
7. y sin 8. 9. 30. 31. sin y cos 1 y sin y ln y tan7 sin 3 33. 34. 4 31 31 3 5 i 0 1 35. y tan i 0 3 i 3 36. 3 1 i 1 1 y i 1 i 9 3 ( 7) 1 3. sin y 1 Theory questions: 1. Which limits are worth computing?. Why is the unction 1 4 limits? not suitable or practice work on 3. Which operation that can be done on a graphing calculator corresponds to the process o computing limits? 4. Can lim c eist when cannot be evaluated at 5. What is the correct notation or each o the one-sided limits o? c? y 1/ e at 6. What is the dierence between a one-sided limit and a two-sided limit? Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 9
Templated questions: 1. Whenever you work with a unction, identiy any values or which it is worth computing the limit.. Compute any non-boring limit or any unction o your choice. What questions do you have or your instructor? Dierential Calculus Chapter 1: Limits and continuity Section 1: The concept o limit Page 10