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Calculus: One-Sided Limits


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:18
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 57 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
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 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 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.

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The Concept of the Limit
One-Sided Limits Page [1 of 2]
Okay, so this idea of a limit, what you're heading toward, the height you're heading toward as the x's get closer and closer to a particular fixed value.
So the way to think about this graphically is to look down and say, "Okay, suppose I want to approach b? If I let the x's approach b, what is the function doing?" So I superimpose the action of the function with my fingers and I ask, "Are my fingers coming together?" And the answer is yes, they're coming together, they want to touch each other. And at what height are they coming together? Well, you can measure, look across there, and see it seems to be right at this height number S. So, in fact, the limit x b of f(x) = S. That's what it means. Great!
What about here at c? Well, as I approach c, I'm going to see now what the function is doing, so I now imagine going up to the function and I head now towards c, and the question is, "Are my fingers coming together?" They absolutely are coming together. They're heading toward each other. What height is that? Well, you can measure that height and see that that height, in fact, is coming in to be T. So, in fact, that limit is T.
Now, of course, you can see there's a hole there, so the function isn't defined there, but remember, with a limit, all I care about is what's going on as I approach this target, not what happens at the target. So you can almost imagine covering this up, putting the window pane right in there, and look around it. And you can see I'm heading toward T. So here I'd say the limit x c of f(x) = T. Well, great!
Now, what about a? So I want to now compute the limit x a of f(x). And what is that? Well, let's see what happens. As I let my fingers come in, I want to ask, when I superimpose up to the curve, are my fingers coming together? So let's go up to here and see. Now, as I get closer and closer to x = a, are my fingers about to touch? The answer is no. In fact, I can see that there's this gap between my left finger and my right finger. So, in fact, they're not about to touch. This limit does not exist. So this is an example, sadly, where the limit does not exist. That's sort of sad.
However, if you look at the picture, you see there's something going on here. In fact, the reason why this limit doesn't exist is because, you see, what my left hand is doing - it's like this expression the left hand doesn't know what the right hand is doing - that's exactly what's going on here. The left hand is coming this way. In fact, if I only use my left hand, you can see I'm actually heading toward the number L. And if I only use my right hand, you can see I'm heading toward the number R. And the problem is that L and R are different. If they were the same, then I would be coming together just like this, like we saw when I approached T.
So, this actually sort of inspires a new idea, the idea of a one-sided limit. So, for example, a left-handed limit is what happens when you approach something just from the left-hand side, just using your left hand. So, in this case, what I can say is that while the limit doesn't exist, I can take a look at the left-handed limit. And what would that look like? Well, first of all, let me show you how we would write this: limit as x a, and then we put a little minus sign to tell us that we're going to be hitting this thing with values that are a little bit smaller than a, which means we're coming in from the left. All these values are a little bit less than a. So it means we're coming in from the left. This is a left-handed limit, and so you only use one hand. You don't ask, "If my fingers are coming together," all you ask is, "Is my finger heading toward a particular value?" It is heading toward a value, it's heading toward a height of L. So, in fact this left-handed limit = L.
Can you figure out what I mean by right-handed limits? I bet you can, because all you do now is use your right hand and you hide your left hand. But if you have a pen in your left hand, don't put it behind your back without putting the cap on it, otherwise your backside will be marred for life. So, if I come in just on the right and approach a, what am I heading towards? Well, I am heading towards something and I'm heading toward a height of R. So, in fact, I'd say that the right hand limit, as x a, and I put a little plus sign there to signify the fact that really what's going on here is I'm getting closer to a, but always with values a little bit bigger than a, a little bit larger than a. And if I see where f(x) heads towards, I see it goes to R.
So the left-hand limit is L, the right-hand limit is R, and since L R, that implies that this limit doesn't exist, because my left hand doesn't know what my right hand is doing. When, in fact, these two numbers are equal, then we say the limit does exist. And you'll notice now that, in all the previous examples that we looked at here, both the left- and the right-hand limits are equal.
For example, let's look at c. The left-hand limit up to c is T and the right-hand limit is T, so therefore the limit exists and it equals T.
Anyway, now you can talk about one-handed limits, if you so desire. And if not, then move on. I'll see you at the next lecture.

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