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Calculus: Evaluating Limits

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About this Lesson

  • Type: Video Tutorial
  • Length: 19:09
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 207 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Limits (12 lessons, $19.80)
Calculus: Evaluating Limits (4 lessons, $8.91)

The limit is the range value that a function is tending towards as you get closer to a domain value. In this lesson you will learn about evaluating limits by substituting values into the expression as well as what notation denotes a limit (and how to read and interpret the notation). If a function is well-behaved, the limit will be equal to the function at that point. Well-behaved functions include lines, parabolas and square root functions. Poorly-behaved functions, however, include piecewise-defined functions and step functions. We will also look at how to evaluate limits by canceling (which is what you should do when you get an answer of 0/0, an indeterminate form, after trying to solve for the limit using substitution). Limits that produce indeterminant forms may or may not exist. An indeterminate form is a mathematically meaningless expression and, in this case, it is just an indication that more work needs to be done to evaluate the limit. Professor Burger will walk you through how to progress once you arrive at an indeterminate form.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

Thinkwell
Thinkwell
2174 lessons
Joined:
11/13/2008

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_blu
Great job!
01/19/2013
~ smgann

Thought the instructor did a great job explaining limits.

9-21-2007-06_homepage
Finally!
02/10/2011
~ JMacy

Limits finally make sense to me! I was having a hard time wondering why we would want to find the limit of a function but thanks to the Doctor here, I finally understand it and appreciate it that much more.

Nopic_grn
AAAAAAAAAAAA++++++++
04/22/2010
~ joel8

Great!!!! Recommended to all!

Jfilip_homepage
Great lesson
08/05/2009
~ jfilip

This guy teaches this stuff awesomely. Even better is I don't feel awkward when I rewind the video 6 times to make sure I understand what he just said. It's like having a tutor who not only won't but can't judge you! Awesome.

Nopic_blu
Great job!
01/19/2013
~ smgann

Thought the instructor did a great job explaining limits.

9-21-2007-06_homepage
Finally!
02/10/2011
~ JMacy

Limits finally make sense to me! I was having a hard time wondering why we would want to find the limit of a function but thanks to the Doctor here, I finally understand it and appreciate it that much more.

Nopic_grn
AAAAAAAAAAAA++++++++
04/22/2010
~ joel8

Great!!!! Recommended to all!

Jfilip_homepage
Great lesson
08/05/2009
~ jfilip

This guy teaches this stuff awesomely. Even better is I don't feel awkward when I rewind the video 6 times to make sure I understand what he just said. It's like having a tutor who not only won't but can't judge you! Awesome.

Limits
Evaluating Limits
Evaluating Limits Page [1 of 4]
Limits – how do we make sense of them? This idea—the sort of touchy-feely idea—of things coming together and asking what things approach seems not very mathematical, even, and it seems not very quantitative. So, how do we actually solidify these ideas?
Before we get going lets just give you an opportunity to actually try your hand at some of these things by looking at some pictures and seeing if you can guess what the limits are. So, try this as sort of a little warm-up before we even get going here.
Well, now that we’ve warmed up a little bit here, lets take a look at some examples together and start to build up some ideas of what’s going on. So, to look a limit, the idea, graphically, again is that all we care about is what is happening as you approach a particular value. So if I have a function like this, let’s say, and I’m going to actually try to indicate that we don’t care what’s going on actually at that value right there; but all we care about is what’s going on around it.
So, if I put, let’s say, a five here—that seems to be the magic number we’ve been using at this point. The question is if this a function f of x, the question I may ask is: What is the function approaching in terms of its height? What is the height—or the y-value—approaching as the x-values get closer and closer to five? And the way I think that’s really a good way of thinking about this is to think about running your fingers along the curve itself and asking yourself, “As I get closer and closer to the x-value of five, are my fingers about to touch?” And if the answer is, “Yes, my fingers are about to touch,” that means that limit will exist, and it will equal the value—whatever that value is where my fingers would touch if I were allowed to just keep going and pass the yellow here.
So, for example, in this case, the limit would exist, and it would equal this value here—whatever that height is in there. That height would be five as well or six or something, okay? So, the idea of a limit is this notion of coming together and seeing what the height approaches as the x-value approaches something.
Now, there’s some notation that we can write for this, and I want to now show you the notation. The notation is -- I would write the limit as x approaches five of f of x. Now, this is very intimidating. And you know why? This is like anything else—like a foreign language. It is a new kind of thing, and so, therefore, it looks so scary. But let me just tell you, this is just language for the exact ideas we’ve been talking about in the previous discussion. In particular, this just means: “I want to know the value that the function is heading toward as the x’s head toward five.” Okay?
So you would read this—by the way, here’s how you read this in English—if you see this in a child’s story—here’s how you read this thing. You would say, “The limit as x approaches five of f of x. The limit as x approaches five of f of x.” That’s how you’d read that. And what does it mean? It asks for the y-value that you’re heading towards as the x’s get closer and closer to five.
Let’s take a look at some examples and see what we get. So, for example, what would be the limit as x approaches three of the function two x plus one? Well, what would the answer to that be? In fact, why don’t you just take a guess right now, if you can. We won’t stop, or anything, but just see if you can just look at this and make a guess, and I don’t care if it’s right or wrong, again. The idea is just to make a guess and see, but maybe you can actually guess it right. Don’t worry about how you got it or anything, but ask yourself, “What is this approaching as x approaches three?”
Well, one way of thinking about this, would be to draw a picture of it, like we’ve always done, and take a look at the graph. This is a straight line, so we can actually graph this. And if we graph it, well, we see the y-intercept is one. They go up one. And then the slope is two. So, that’s a slope—change in y over change in x—so that’s for every two units in the y, I go one unit into the x. So I go one over in the x and then two units up—one, two. One over, two up, one over, two up, one over, two up. Now if I connect these points, I get the straight line. So, now as they get closer and closer to three, are my fingers coming together? Well, you go up to the line here, you see? And you see, indeed, my fingers are getting closer and closer together. If the limit does exist, there will be an answer. And what is it? Well, where would it be? Well, it actually would be where that dot is. Not a very interesting example because, in fact, I’m heading toward what the function actually equals. And what value is that? Well, if you would reach over here and read it, you would just plug in the three, and you would see it would be seven.
So, maybe you guessed seven. Maybe you didn’t. Either way, I’m happy as long as you guessed something. But the answer would be seven, because as you get closer and closer to three, the y-value’s get closer and closer to this value, right here, which if you would actually evaluate this, you’d see a three in here—two times three plus one is seven.
In fact, this simple example indicates something very, very important. If the function is a very, very nice function in some sense—if the function is very, very well-behaved—and you can just plug in the x-value three into the function and get the answer of seven—you know what? That’s the end of the story. It’s actually pretty easy. Because those are the examples where, in fact, the thing that the function is heading towards, is exactly equal to what the function is at the point. There’s no hole. There’s no jump. There’s no craziness. It’s sort of a nice, friendly example. In fact, what you’re heading towards is indeed, what you will equal. And so if you can just plug in and get the answer, it’s real easy.
So, in fact, limits are really easy for these kinds of examples. Let’s take a look at some other example. For example, let’s take a look at this one. The limit as x approaches minus two of x squared plus x. Okay, now first of all, let’s just understand what this notation means. I’d read it as, “The limit as x approaches minus two of x squared plus x.” What does that mean? Well, this is a function—function x squared plus x. And my question to you is, “Well, what is this function approaching as the x’s get closer and closer to negative two.”
Okay. Well, how would you find that? We could graph that. That’s a parabola—a happy face, a sad face. Happy face. And so, we could take a look and see what’s going on at negative two. If we would insert negative two in here, by the way, and what would we see for x? If we actually plug in negative two, we’d see negative two squared, which is a positive four and then plus negative two. So we’d see a positive four plus negative two, which would give me two. Well, since this thing is well behaved—it’s nice. There’s no holes there; in fact, the limit is two. What we’re approaching is the same thing as what the function’s defined to be there. It’s a parabola. There are no holes. There’s no nooks, there’s no crannies. It’s actually pretty simple.
And you might be saying at this point—and boy, this would great if you’re saying this—I’d be really excited—say, “Limits are a piece of cake.” If only that were true, if only that were true. But they are a piece of cake if you can plug in, get an answer, and you’re set.
Let’s try another example. Limit as x approaches four of the square root of x plus three x minus one all divided by x minus three. Look at that one. Now, that one is looking really complicated. There’s square roots in there. There’s a division. This is probably going to be a real hard one. But before I try anything else, and before I panic, what I’m going to see is what happens at four. What is this thing doing at four? Well, let’s plug in x equals four into this function right here. If I were to plug in x equals four, I would see the square root of four plus three times four minus one all divided by four minus three. And what does that equal? Well, that equals—well, the square root of four is two. And so, I have two plus twelve minus one over four minus three is one. So, I see here is twelve plus two, which is fourteen, minus one is thirteen, so actually thirteen over one, thirteen over one. Well, that’s a number. And in fact, that tells me that this function is actually pretty well behaved around four and the limit, therefore, is going to be equal to thirteen.
So, even though this looks sort of complicated, it wasn’t complicated, because near four, it turns out whatever this looks like – and notice something interesting here, folks. I don’t know what this looks like at this moment. It’s some complicated picture, which I honestly don’t know. But it turns out that math is allowing me to tell that as I approach four, the value is approaching thirteen, even though I don’t know what the picture looks like exactly, so I can’t even draw it. So now, in some sense, we have to let those drawings fade into the background of our reality in mathematics. But instead, just are kept as the germ of the idea of what’s going on, but now we’re actually able to—we’re empowered to actually do more difficult, challenging functions.
Okay, but so far, again, this technique of just plugging in seems to work. So let’s look at some examples where interesting things happen. So now let’s take a look at the following example. Limit as x approaches zero of x cubed plus x, all divided by x. This is a nice example. Let’s think about that for a second. Well, let’s see. I’m letting x approach zero and I want to see what’s this thing’s approaching. So, what I do is I go off and I’m going to now plug in the zero, like we did before. Always try that first. That’s what I always do. So what do I get? Well, I see zero cubed plus zero, all divided by zero -- and that is zero over zero. So, if I plug in, I get zero over zero and as we all know, this is bad news, bad news.
So, in fact, what does this mean? Does it mean the limit doesn’t exist? Does it mean we can’t do the problem? Does it mean…I don’t know what. The answer is this means that the simple and naïve idea of plugging in the zero here is not going to work. And when you get a limit that when you plug in, produces a zero over zero, there’s a fancy name for that in the math world, because it really scares us -- shouldn’t just scare you as sort of a math fan, math enthusiast, amateur, but it actually scares the professionals – and we call this, an indeterminate form, an indeterminate form.
When you take a limit, and you end up—when you try to plug in—with a zero over zero, that’s an indeterminate form. Does that mean the limit doesn’t exist? Absolutely not! It means it’s an indeterminate form. It means it needs more work. And that is so important, I want to write that out for you, live. Needs more work.
Now, you know, a lot of times what students think—this is a classic, classic mistake, by the way, and one that I am sure that you’re going to make, not because you’re having difficulty, but just because everyone makes mistakes. Everyone makes this mistake. You’re going to do a problem one day—I just hope it’s not on the quiz—and you’re going to plug in, and you’re going to get zero over zero and you’re going to say, “That limit doesn’t exist.” Okay? That is not right, necessarily. The answer is if you plug in and get zero over zero, it’s an indeterminate form. That means that it needs more work. The naïve approach is not going to work, because what this is telling us is that the function, at that point, is undefined. But it may be a hole, so you actually still be heading somewhere, or maybe something really crazy, and maybe it’s not heading anywhere. We just don’t know. It’s still a mystery.
So how would you resolve this mystery? Well, one way to resolve the mystery is to remember one key fact. Remember the window story. What are we doing here? Is this the window shutter or is this the window with the frame? Remember, it’s the limit, so we’re curious about what’s going on around but not at the point zero. So this is a window frame, where there’s that frame right there, blocking the view. You can’t quite see me, because I’m being blocked. You can see everything here, and you can see everything here, but you can’t see what’s going on there. Okay?
Well, if that’s the situation, then, this actually isn’t so bad, because you might look at this and say, “Gee, I wonder if I could do a little bit of Algebra here?” Notice that in fact you could immediately factor out something on the top. We can factor out an x. So, if I factor out an x, then I can actually produce something and what I’d like to do is show you exactly how that would look right now. So, I’m going to try that. Let’s write this down. The limit as x approaches zero of x cubed plus x over x. Now I want to factor out the common factor of x on the top.
Now, when I do that, I’m going to write an equal sign here and this is an important point. You know -- I don’t know maybe the person who you’re working with, learning about Calculus in real time, won’t really care about this but boy, I care about this. If you write an equal sign, you’ve got to keep writing the limit for me to remind me we have to take the limit. Boy, my students always get in trouble if they forget to write down this limit thing. So, aren’t you thankful you’re not one of my students? Yes, I know you are, but please write this anyway. I think it’s a good habit to get into.
So factor out the x, and then I get an x squared plus one all, divided by x. And now you look at this, and I want you to see something really neat here. There’s something really fun happening here. Ideas are being drawn together, because when you look at this, you can now see how we got zero over zero when we plugged in. Do you see the zero over zero? It’s not that piece. That piece is sort of a harmless piece. There’s the culprit of zero on the top. There’s the culprit of zero on the bottom. That gave you the zero over zero. You can see the zero over zero right there. That was why we got this to be an indeterminate form.
You see it? You can really feel it now. But the happy thing is since that’s the multiple of the top, and I’ve got an x downstairs, I can happily cancel away. Now I can cancel away with one proviso. Remember the proviso, when you want to cancel, you’ve got to make a promise to me. What’s the promise you have to make? The promise you have to make is that whatever you’re canceling is never zero. So you have to promise me that x does not equal zero. Is that a problem in this question? Remember the windowpane. All I care about is what’s going on as I get really, really, really close to zero. But do I care about what’s going on at zero, at the equal zero moment? The answer is no. That’s being blocked by the pane. Right? Remember that? It’s being blocked. So, I don’t care what’s happening at the moment it equals zero. All I care about is what’s happening around zero.
So, in fact, this little proviso thing is perfect for the question at hand. You see that? Because I’m allowed to do that, because, in fact, x is not going to equal zero. X is only going to get close to zero. Do you see this? So this is wonderful. So, we made progress. That’s completely okay. No need to fret about that.
And now what do I do? Well, now I write equals, and I have the limit as x approaches zero. And now I’m just left with this x squared plus one. Well, now this limit is actually a pretty easy one. It’s like the ones we’ve seen before. Here, this thing, it’s okay. I can just plug in the zero and I actually get a number. I would get zero squared plus one, which would equal one.
So, in fact, this limit turns out to equal one. Remember, originally, it gave us zero over zero. So this is an example of an indeterminate form. It needed more work. In this case, the work that we did was we factored out an x, cancelled, and we got this. So, it turns out this limit is one. This limit is one, one!
So, now, how can you actually think about this result. I mean, it’s sort of abstract. The limit is one. Well, it turns out, we actually have seen this function before in our travels. In fact, maybe you remember, maybe not, but when we were looking at graphs of quadratics and other sort of simple curves, we actually looked at the graph of that. That was for the foreshadow, and I actually planned this out in advance, believe it or not.
Let me remind you what that graph is, and I’m going to show it right over here. Take a look. Remember the graph of that? We saw that earlier and noticed that, in fact, there is that hole at x equals zero, but it’s just a hole. And you’ll notice that as you approach that value, x equals zero, on the curve, you’re coming down that parabola. You’re coming down, and you are heading towards something. Your fingers want to touch as they come down that parabola and go in right there. You see it? They want to touch each other. And where do they want to touch? Well, what’s the y-value there? You can see very instantly the y-value there is one. That’s exactly the one we got here.
So, in fact, here’s an example where you can actually visually see the hole. We already saw this example earlier. But, now, we also algebraically, by taking these moves, factoring out the x, canceling it away with this condition, which for a limit is no problem. Because, remember that we only approach zero, we never equal zero, and we just happily go along. Now here’s an example of an indeterminate form where we got zero over zero, a little more work, and we actually got an answer.
All right, how about now you try your hand at one of these things, and what we’re going to do next, by the way, is just a lot of examples. We’re going to have a lot of fun with this stuff to really nail this stuff down. Not hard once you get the feeling for it. A little shaky until we iron out the kinks. Okay, see you in a bit.

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