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Calculus: Introduction to L'Hopital's Rule


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:45
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 83 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: L'Hopital's Rule (8 lessons, $11.88)
Calculus: Indeterminate Quotients (4 lessons, $6.93)

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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L'Hospital's Rule
Indeterminate Quotients
An Intro to L'Hospital's Rule Page [1 of 2]
So, there's indeterminate forms, when you see 0/0 when you're taking the limit or infinity over infinity when you're taking the limit and just trying to plug in the x value. How do you tackle those things? Well, we saw two ways of tackling those things. The first way is, if you can factor or do some sort of algebraic gymnastics you might be able to simplify it. Sort of isolate the 0/0 term, cancel it away and then you get the answer. Another technique is just to try to get rid of all the high powers of x by dividing top and bottom by x to the highest power. Then you'll see a lot of 0 terms and then potentially some other things. Then you can see what the answer is. In this case, the answer doesn't exist or you might want to say equals negative infinity.
What about more exotic functions? Is there a method that will allow us to see what's going on without actually doing some of these gymnastics? The answer is happily, yes. The method is called L'Hospital's rule. So, L'Hospital's rule actually tames these indeterminate forms. Those question marks become exclamation points. So, how do you actually do it? Well, the idea is really neat. Basically, here's the concept. The concept is if you're looking at some function that has a top and a bottom and you're taking the limit and you get an indeterminate form, so you see 0/0 or see infinity over infinity. Suppose that you can actually take the derivative of the top. Suppose that you can take the derivative of the bottom. Then if you can do that, then the limit, that answer will be the exact same answer as taking the same limit, but now of the derivative of the top divided by the derivative of the bottom.
Now that seems a little peculiar and it seems like why would you even want to do that? But remember when you're taking derivatives of polynomials, the degrees of the polynomials get smaller. So in some sense we're taking a hard problem and converting it to an easier problem that might be able to be solved. So, that's sort of the strategy. Now let me show L'Hospital's rule in action. In fact, let's return back to these examples that we took a look at. So, here's the first example. Well, we already figure out the answer by factoring. Well, let's see if we can use L'Hospital's rule to see what the answer would be.
Now, I see I'm getting indeterminate form. That's the first check. I see 0/0 when I try to let x approach 3. So, what do I do? Well, what I do is I take the derivative of the top and take the derivative of the bottom and I consider a new limit. So, the new limit I'm going to now consider is the following. L'Hospital's rule says that these things are equal. So, now watch this. It's like magic. It's the limit as x * 3, but now I'm going to change this and I'm going to take the derivative of the top. The derivative of the top, that's easy to do. It's just 4x - 0, so 4x and the derivative of the bottom, is happily, just 1. Now, I take that limit. Well, that limit is really easy. If I let x approach 3, I just see 12/1, which is 12. Which is what we got before. Isn't that great? It's like magic. This always works when you can take derivatives.
So, what are the key steps? The key steps are many fold. First of all, you have to make sure you have an indeterminate form. So, you have to make sure that you're either seeing 0/0, so you want to see that 0/0 or you want to see that infinity over infinity. So that's the first thing you need to see. Then, if you can take the derivative of the top, take the derivative of the bottom, then this limit will be equal to this different looking limit. The answer will be the same and this time you might be able to actually compute the limit. So, that's pretty neat.
Now, so let's try another example. In fact, let's review this example that we say earlier here. This is the limit where we got infinity over infinity. Now what if we use L'Hospital's rule again? Are we going to get the same answer? I hope so, otherwise we're in big trouble. By the way, you know you might be saying, "Who is L'Hospital?" Right? This is a guy from the 1690's. This was a guy who was in Europe and he really was a math fan, but he wasn't really a mathematician. He was a math fan, like you. You probably love math. Right? Don't answer that question. He really was a math fan. He was a rich guy. He was a rich guy and he would sort of throw money around. He was sort of like a big shot, if there were cars he would be driving a really fancy Mercedes or something. There were no cars back then, so he just did math instead.
He just loved thinking about math. In fact, he loved it so much he would hire people to think about math. He was writing a book, like a textbook. He was writing a book and he just want to do math, math, math. So, he hired a guy named John Bernoulli. Now John Bernoulli, he's a big mathematician, there's Bernoulli equations, there's Bernoulli this. You can't spit in math without hitting a Bernoulli. L'Hospital hired this guy to do some math for him and it was actually Bernoulli who came up with this really neat idea. But then L'Hospital put it in his book in like 1692 or something and now it's called L'Hospital's rule. So, bottom line, moral, if you want to become famous in math all you've got to do is pay a really good math person to do some math and then you just sort of farm it out as your own.
So, in fact, I'm for hire.
Anyway, let's try this one. Now, here I see, again, an infinity over infinity type thing. So, what do I do? Well, I can differentiate the top and I can differentiate the bottom because those are just polynomials. So, I take the limit as x approaches infinity and if I take the derivative of the top I see -3x² + 9 - 0 divided by--and the derivative of the bottom is just 20x + 0. Okay, now let's let x race off to infinity. What happens? Well, on the top I'm still approaching negative infinity. This is still getting really, really big with a negative sign in front. On the bottom I'm still approaching infinity. So this is like an infinity over infinity, a negative infinity over infinity, whatever. It's still humungous over humungous. So, are we done? No. This is still, this is still an indeterminate form.
So what would we do in this case? Well, if you've got a thing that works, use it again. So, I'll just apply L'Hospital's rule one more time to this new thing. So let's see what happens. Here's the original person. This is the modified person, but we're still not home free. I'll apply L'Hospital's rule one last time or maybe I'll have to apply it more times. Who knows? We'll see. Well, if I take the derivative of this I see -6x + 0 and on the bottom, the derivative of the bottom is just 20. Now I see immediately I'm not going to get either 0/0 or infinity over infinity, because that's just 20 on the bottom. That's a constant. So what's this thing approaching? Well, as x gets arbitrarily large the top is going to negative infinity, but the bottom is just staying at 20. So, in fact, this whole thing either doesn't exist or it equals negative infinity. Look, that's what we got before. This is great all you have to do is take derivatives.
Now, I want to say one little, teeny word of caution to you. Now this is really important. You know I know you're on the ball and you're thinking, "Gee, if I'm taking a derivative of a quotient I should use the quotient rule." You're absolutely right. The L'Hospital's rule doesn't say, "Take the derivative of the quotient." This sneaky technique works in the following way. What you do is, if you have a limit that's an indeterminate form, that will a new limit where you take just the derivative of the top and formally divide it by the derivative of the bottom. I don't want you to think that you're taking the derivative of the whole quotient, because that requires the quotient rule. No. What we're doing here is a very formal think, derivative of the top divide by derivative of the bottom and then we take the limit. Okay, we'll take a look at some more even exotic examples coming up next. I'll see you there.

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