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About this Lesson
 Type: Video Tutorial
 Length: 14:40
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 159 MB
 Posted: 06/26/2009
This lesson is part of the following series:
Calculus (279 lessons, $198.00)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Limits (12 lessons, $19.80)
Calculus: The Concept of the Limit (8 lessons, $12.87)
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. 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'Hopital'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 UTAustin 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, padic 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
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Limits
The Concept of the Limit
Finding Limits Graphically Page [1 of 3]
Okay. So, now we're at the point where we really want to get the sense of how we can mathematically bring things closer and closer and closer together. And to really think about this issue, I want us to read them, and one of the things we talked about earlier, and that is looking at pictures, looking at graphs of functions. Now, this graph over here which I've set up, you can see the axes and I've labeled one, two, three, four, all the way up to ten, and one to nine here, and then I have a graph of a function here. And that's how we're used to looking at these things. We're used to looking at these things by looking at the whole graph. It's very pretty. This is in blue; for example, it can be in different colors. And you look at the graph.
What I would like for us to do now is to actually take a look at this picture, the picture that we've been looking at since we were kids, but now with a different eye. In particular, I want us to look at this in two different new ways. And the way I want to sort of inspire this new way of thinking is to think about looking at the graph through a window. Okay? And there's two ways of looking through a graph, looking at the graph through a window. And I want to show you those two ways right now. They are very different and, believe it or not, this is really what's at the heart of what we're doing here. So, so watch this.
The first one, I could put a window in, looks like this. Now, that's just a window that, that you could see there's a window. There's a window, uh, I guess it's not really paned, but you can that there's a wood frame around it. There is a frame here. You have a very good view of the function. In fact, you have almost a complete view of the function, looking out, gazing through the window to the function. Uh, notice that there is actually one point, though, that's missing, in fact, let me emphasize that a little bit. One point that's missing. In fact, the action at five is actually blocked because of the frame of the window. You see that the frame of the window actually blocks, so you don't actually know what's going on when X equals five. You don't know where the functions, because it's blocked there, but outside of that you get a beautiful picture of everything else. Okay? So that's one view that I want us to think about.
The other view is a similar view, let me show you the other view. The other view is actually just a view through a window with, uh, with shutters. Look at that. Now, here what we see is, me just waking up in the morning, for example, waking up in the morning, these are closed. You can't see really anything, uh, except you can see, notice there's a little space. I hope you can see that, there's a little space between those two shutters. And if you look really, really closely you can see that, you can see where the function is at that one point. At five, you can go up and see what the function is. So, here you can see what's going on at five, but you don't know what's going on anywhere else because you just woke up. But at five, you can see. If you wanted to see the whole thing you'd have to actually open up the shutters, look, and there's the function again.
Now I want you to think about the two views of looking at a graph. One view is looking at the graph, just waking up in the morning, through closed shutters where there's a little slit, but then you can only see one point. You just know that point and nothing else, and then the other view, the more interesting view, I think, because that's the view where you see every single thing except what's going on at that point. There's a difference here. First of all, you can see it. Here you see all this detail, but you don't know what's going on there, that's, that's covered up, it's a mystery. Here you can see the beautiful detail of my window, the beautiful detail of my, uh, of my, of my sha, of my, uh, shutters, but you can't see the function. All you can see is that one point right there, that one point right there. These are two different views and, believe it or not, these are the distinctions that are going to lead us into calculus. This view is the view we've been looking at, believe it or not, all our lives. In fact, we've basically been waking up to mathematics all up to this point, until this moment where we're finally going to open this up. Because what this view is telling us is what's going on at five. If this function is like F of X, this is saying this height Y equals F of five. So this view is really the F of five view. What is the value of the function at five? You plug in five, you do some stuff, and you spit out the answer, and that, in this case, looks like it's going to be around six or so. That's what we've been doing up to this point.
Now, what we're going to be doing which is new is looking at this view, and this is the calculus view. The calculus view is where you don't know what F of five is, but you know everything else around it. You know everything else around it. Now you may be think, thinking, but what's the big deal. If I know everything else around it, I can just see where it's going, it's going to, you know, six. What's the big deal? I don't see what the problem is. Well, I'm not saying there's a problem. I'm saying there's two different views.
But, now, what I want to do is illustrate the distinctions between these views with some more interesting examples than just this very pretty, but simple function. So, let's take a look at this example. Now, with this example I have a different function that's being covered up by the frame of the window, but what's going on at five. Well, you could take a guess, probably. All right, at five it looks like, that's around what between five and six, maybe five and a half, is your guess. Well, let's see what the actual function is doing. Whoa! Now, what's that all about? Well, it turns out that the actual function  where's the dot? If I use this view, it tells me the value of the function there is actually around four. But, when you look at it with this view, it looked like the answer was going to be around six. And what's the distinction? The distinction is that this function was not nearly as well behaved as the first one. This actually has a little place where we stopped. There's a little hole there, there's a little hole right there, and there's a jump. The function drops down, drops a point, jumps back up and keeps going. So, in fact, you can see that these two views actually give two different answers. If you look at F of five, the answer there is going to be four. But if we ask, what does this view show us? That view, which is saying what is the function heading toward. The function is heading toward a value of around five or six. There's a difference in these views.
Let's take a look at another example. Take a look at this one. Now this one, you can see the dot right there, there's the point. So what is F of five, F of five looks like it's around seven. F of five is seven, and so, what's the function heading towards? Well, it's heading towards seven, but let's open up the window and see what it actually is doing. Whoa! Look at that! Actually, the function is way down here somewhere. There's a big break in this function and there's a lot of chaotic things going on here because what's actually going on is there's sort of a, a hole over here, but the function is heading this way. Heading to what, in fact, let me move these shutters out of the way. The function is heading up toward a height of around, oh, I'd say five or so, and heading this way, it's heading down to around four or so. And then the actual value of the function is around seven. So, what's going on here? This view completely obscured the nuances of the actual function, where as this view would have told us something very strange is going on.
See the difference in these views? Let's look at some more examples. In fact, let's look at this one. Let's look at this view. Now, what about this view? What's going on at five? Well, if you look at this, in fact, let me just, just squeeze this over so it really is at five, so, there we go, okay. What's going on at five? Well, if you look at this picture your guess might be, uh, the action is around, oh, seven, but look at the reality of things. It's not defined at all. It's a hole. We've seen holes like this when we were looking at functions before. This is just a big old hole here. The function's not defined at all. Yet, you can tell me where the function wants to go, where the function is heading towards. Notice that as these pins get closer and closer together, it turns out I'm approaching this point seven. Is the function defined there? No, but I can tell you, looking at this window, that it wants to go to seven. It wants to go to seven.
And this is the new idea. This is the new wrinkle. And this idea is the question of I don't care what's going on at exactly five. As for five, that was good for us when we were kids, but now, I'm not interested in that. What I am interested in, is what's going on around five. I don't want to think anymore about waking up in the morning with these shutters. I want to open the shutters and take a look out the window, and the only deal here is that now that window has that frame going right through it. So, I don't know what's going on at five, and guess what? I don't care anymore what's going on at five. I'm interested in the more interesting question; I'm interested in what's going on around five.
What is this function heading towards as I get closer and closer to five? And, so, you can see this, in this example, for example, it's not even defined; yet you can see what things are heading towards. Now, how could you actually sort of feel this or see this? Well, one way of thinking about this is actually taking your fingers and letting them ride along the surface of the, of the function, and ask yourself as you get closer and closer to the value X equals five, are your fingers wanting to come together. And in this example, you can see the answer is yes. In fact, I wanted a really fancy visual prop for this, you know, this is sort of high tech web stuff, right? But that didn't come, maybe some kind of things coming in, and, you know, so this is what they came up with. Very high tech, they said, "We'll give you a hand." Well, great. So, I got, now I've got two hands in fact. So this is actually the way to think about this idea. Let your fingers or some else's fingers if you can afford them. Let your fingers actually drift up along the function. Now, you're going to let the, the, the X value get closer and closer to five. Don't touch five; just get closer and closer. And the question is as you get closer and closer to five along the function, are your fingers coming toward each other? In this case, the answer is yes.
And the next question is okay, what is the height that they are coming towards? What's the height that they want to meet at? I don't care what happens at the point. And so, you put back the, the window shade and you sort of run along there, and what do you see? They are getting closer and closer to each other and that height right there, you can measure it, is, indeed, around seven. So their height is around seven; therefore, I'd say that this function is approaching seven as we approach the number five. Did you get that?
Let's try another example. Here's another one. In fact, let me just move this out of the way here, let me put up another example. This one we'll just put up here like this. Oops, I'm sorry, put this right here. Here's another example. And my question now to you is well; first, we're not going to ask the old fashioned question, yuck! What is F of five, yawn  F of five. What do you do? You go to five, and you look up and see where the dot is. So you could use the old, the old window shade thing, or you could just not see what's going on. I just care about what's going on at five. I go over and I see around seven. So, at the five is seven. Big deal.
I'm now interested in the more interesting thing, the question of what's the function tending towards? What's the function heading towards? What does the function want to go to? And the answer here would be, again, putting this window shade, not to distract you from the actual point, but now let's sort of try to wean yourself off that. And just look at this and think to yourself, "Well, as I get closer and closer to five and I look up into the function, what is the function getting closer and closer towards." So first of all, are my fingers coming together? Are your fingers coming together? Absolutely, they are coming together; they're about to touch. They're about to touch. And where are they about to touch? They're about to touch at seven? No, they're not about to touch at seven. Seven is light years away. They're about to touch over here, at around five. So, in fact, here the function wants to head toward, it's tending toward five. Does everyone see that? The only thing that matters, don't look at what's going on exactly at five, ask what's happening as you get closer and closer to five. My fingers are coming together, they want to touch, and they want to touch at five. What happens at five, we don't care about. So, again, if that's a little bit confusing, just drop back your, your window frame. Drop back your window frame so you're not distracted by that and now you can see it much easier what's going on there, I'm heading toward five. Everyone see it?
Let's try another example. This other example, so these are going to be more scary. Oh, that's more scary because I think I probably dropped it already. Goodness me! Did you miss me? This example we saw earlier. Let's see what the, what the limit is here. Again, you could be distracted by a lot of things. What is F of five? Well, all you do is look at where it's colored in, looks like it's eight. But, now I ask you, what is the function approaching as the X gets closer and closer to five? So, you can cover up that point if it distracts you, and now bring your fingers along, and ask yourself, as you get closer and closer to X equals five, are my fingers coming together? Are they about to touch? No, they are not about to touch. They're light years away. This one's way up here, and this one's way down here. They're not about to touch at all. For them to touch they'd have to be sort of about to touch like that, dink. They're very, very far away. So, in fact, here the function isn't approaching one particular number. My fingers aren't coming together. So, a lot different things can actually happen. And looking at things, and looking at functions in this manner, and asking the question are my fingers coming together? In this case, they're not. Or in this example, where they are, are my fingers coming together, and the answer is, yes they are. They're about to touch and they're about to touch here around seven. This actually is a mathematical notion and the mathematical notion is called a limit. And a limit is just the value that the function is heading towards, or the function is tending to as you get closer and closer to a particular X value.
Okay, so that, I hope, gives you a sense of what's coming up next. We want to actually, remember, we want to inch our way up and now I'm going actually going to show you visually how you can inch your way up. And inching your way up doesn't tell you anything about what's going on at that point. It just tells you what's going on everywhere else but at that point. Okay, so next up, we're going to actually take a look at some simple examples where we can actually compute mathematical limits and then we're going to take a look at some more interesting examples. Okay, so stay with us. See you in a bit.
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