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Calculus: L'Hopital's Rule, 1 to an Infinite Power


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:45
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 126 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: L'Hopital's Rule (8 lessons, $11.88)
Calculus: Other Indeterminate Forms (4 lessons, $6.93)

The limit of a function is called an indeterminate form when it produces a mathematically meaningless result. Some advanced indeterminate forms have to be manipulated before L'Hôpital's rule can be applied to them. In this lesson, you will learn several ways in which logarithmic functions in indeterminate forms can be camouflaged. In order to use L'Hôpital's rule on a camouflaged indeterminate form (e.g. 1^infinity or infinity * 0), you will need to apply logarithmic properties such that you can rewrite the exponential function as a logarithm and convert it into a standard indeterminate form. L'Hopital's rule is applicable to f(x) and g(x) if they are differentiable functions across an interval containing c, except possibly at c. If the limit as x approaches c of f(x)/g(x) produces the indeterminant form of 0/0 or (+-infinity)/(+-infinity), then the limit as x approaches c or f(x)/g(x) is equal to the limit as x approaches c of [the derivative of f(x)] / [the derivative of g(x)] provided the limit on the right exists or is infinite. L'Hôpital's rule can also be applied to one-sided limits and, as long as the limit is indeterminate, you can take the limit again if L'Hôpital's rule doesn't give you an answer the first time around (and you are again left with an indeterminate form as the limit)

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 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

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Recent Reviews

Thought the video was great
~ Stephen22

Very helpful for me as I plunge forward in Calc 2. For some reason I was making a minor mistake differentiating the ln(f(x)) portion of this problem. Good to always make sure you are placing items correctly in the numerator or denominator. I was working on a similar problem that was the same except it was 3/x instead of 1/x. Of course the answer is still e! Thanks!

Thought the video was great
~ Stephen22

Very helpful for me as I plunge forward in Calc 2. For some reason I was making a minor mistake differentiating the ln(f(x)) portion of this problem. Good to always make sure you are placing items correctly in the numerator or denominator. I was working on a similar problem that was the same except it was 3/x instead of 1/x. Of course the answer is still e! Thanks!

L’Hospital’s Rule
Other Indeterminate Forms
L’Hospital’s Rule and One to the Infinite Power Page [1 of 2]
Okay, now we’re going to do some really cool camouflaged indeterminate forms, and these are great. And, in fact, the first one is genuinely a calculus classic. This is really, really a big deal.
Let me just write down the first indeterminate form I want us to ponder. It’s the limit, as x approaches infinity, of 1 + – now, not a big deal, not a big deal is it? If I were to stop prematurely, not a big deal, because x is approaching infinity, so this whole thing is going to 0. This is 1 over a really, really huge, huge, bigger, bigger thing, so this is going down to 0. So this is just 1, so not a big deal. So what’s the big deal? Here’s the big deal; I'm going to raise all that the to x power. Now, what’s going on here? Well, what I see here is a 1 raised to a really, really big power. That’s sort of like 1?. This is actually an indeterminate form. And it’s an indeterminate form of the sort of garden variety 1?. What is this kind of thing? Well, on the one hand you could say that 1 to the anything is anything, so it should be 1. On the other hand, well, this thing is actually technically a little bit bigger than 1. This is getting smaller and smaller and smaller, but it’s always there. So this is a number a little bit bigger than 1. And if you take a number bigger than 1, like 2 or something, and raise it to higher and higher powers, it goes off to infinity. So, on the one hand, you're saying, “Oh, it should be infinite.” On the other hand, you’re saying, “It should be 1,” an indeterminate form, an indeterminate form. When you're just not sure, then it’s an indeterminate form.
Okay, so what would the answer be? Well, you need to sort of massage this again. Now, what kind of massaging methods can you use? Well, there’s a great, great trick that is extremely useful when you see stuff with x’s in it raised to powers with x’s in it. So when you see x stuff raised to powers with x stuff, here is the lesson for you. And I'm going to write it here in purple, to really emphasize the regalness of this. And it’s a fact that says the following: if you take e and raise it to the natural log of anything, anything, then you know what that equals? That equals just the anything. That equals just the anything. So what I mean by that is the following: if you have eln of 17, that’s going to equal 17, because the e and then this power with a log, they sort of cancel each other out. And if they cancel each other out, you're just left with the anything. So this is a technique – now, why in the world would you want to do this? Well, because there’s this great property of logs – and this is the key to everything, by the way, the key to all of life’s logs is the following. If you have the natural log of something, let me just call it A, and you raise it to some power B, then you know what that equals? That B, which is an exponent, can come out in front as a coefficient. So this is one of the most important and cool facts about the natural log, is that, in fact, it equals B ln A. If you were just going to remember one great fact about natural logs, my first reaction is remember a couple of basic facts about natural logs. But if you really say, “No, no, no, Ed, it’s only going to be one,” I think I’d pick this one. I’d say the fact that if you take the ln AB, that’s always the same as B ln A. If you combine this fact with this fact, you can conquer any indeterminate form of this flavor. Let me show you how you do that.
So the first step is to say, “Well, look, I've got some really complicated thing here.” So I can make that complicated thing even more complicated by writing it as eln of junk. So that’s the first step. So the first step is to write this as e – now watch this. This is going to get really ugly, so if you just ate a meal or something, this might make you a little queasy, but just stay the course or watch it on an empty stomach. eln of all of that junk, so . Look how I'm using this fact. Do you see that? Do you see eln of all that junk; that just equals all that junk. So these two things are the same. This is much more complicated, but they are the same. They are the same, because the e with this as a power and the natural log in the power cancel each other out, and you're just left with the anything. So, in fact, that thing, eln of that thing, is just that thing.
Okay, now why would I do that stuff? Well, because of this fact. See, the thing that makes this problem so hard is that variable in the exponent. But once I've got a log in front, exponents become coefficients. So I can no take that x and migrate it all the way out in front of the natural log, and that’s the power of this particular idea. It allows me to take this exponent, which is going to infinity, and convert it to a multiple. And so now, I use this property, bring the x out in front, and what do I see? What I see is – now, you’ve got a lot of stuff to write and it looks a little scary, I admit. You think I don’t know that? x times the natural log of 1 plus 1 over x. So that’s where we are. And I'm going to now pause here for a second, because what I want to do is – notice that if I want to take this limit as x goes to infinity, all I've got to do is figure out what that approaches. If I knew what that number approaches, then my answer would be e to that number. If this thing ends up approaching 17, then this whole limit is e17. If this whole thing approaches 3, then the answer would be e3. So really all I have to do is look at that limit question and then remember at the very, very end to put it back as e to that power. So now, I’m going to pause and consider a sort of subquestion, namely, what’s the limit just of this exponent? And then I’ll come back later and make sure I put it in the exponent of e. So let’s take a look at that now.
So this is a multistep process. This is not a step for a monosyllabic kind of person. This requires a lot of words and lot of writing, too, by the way. So let me just write up now what I'm going to consider. This is now a different thing. This is not equal. Notice there’s no equal sign. This doesn’t equal that limit. This is a separate little sublimit. So x ln (1 + 1/x)– now, what about that limit? That’s in the exponent. Well, this approaches what? Well, this approaches infinity, so there’s the infinity part. And what about this? Well, now as x approaches infinity, this piece, 1 over that, approaches 0 and so the ln 1 is actually 0. So the natural log of 1 = 0. So this is one of the ? × 0 indeterminate forms. So this is an indeterminate form not of the exact variety we like, but a familiar one. What I want to do now is convert this to a 0/0 or ?/?.
So how do I do that? Well, I'm going to do a double flip. So one double flip, I write the x as 1/(1/x). So I could write it this way, . So let’s just check and make sure that no slight of hand has gone all here. This is all legal. Look, if I've got a 1/x, when I invert and multiply, watch what happens to the x. It comes back on top. So, in fact, these two things are equal. But now, what happens when I take the limit? On the top I still have the natural log of 1, which is 0, but on the bottom what do I have? On the bottom I have something, 1 over something really big, that’s 0. So now I’ve got 0/0. Finally, we can bring back L’Hospital to save the day, hopefully. So what I'm going to do now is use L’Hôspital’s Rule, take the derivative of the top, the derivative of the bottom and see what happens. You see how each step itself is not that big of a deal, but there are a lot of steps and you’ve got to be really careful. And the potential for making a little teeny typo is very, very great.
So first we take the derivative of that. That is a chain rule. I’ve got natural log of blop, so what’s the derivative of natural log of blop? It’s 1/BLOP. So 1/BLOP, which is , and then I've got to multiply that by the derivative of the inside, the derivative of the blop. The derivative of this is just 1. The derivative of 1 = 0, and the derivative of 1/x, that’s x-1, so the derivative is –1/x2. Why is that? x-1, the derivative would be –x-2. So –x-2 is –1/x2. All right, what about the bottom? Taking the derivative of the bottom, we just did that again. The derivative of 1/x is –1/x2. And happily we see a wonderful occurrence, these minus 1/x2’s just cancel. So that’s fantastic, and so what do I see? I see that this equals, well, the limit, as x approaches infinity of 1/1 + 1/x. Well, what’s that? Well, this is heading to 0. As x gets really, really big, 1/x goes to 0 and so all I'm left with is 1/1. So, in fact, this limit equals 1. The limit equals 1.
Now, does that mean the limit in the original question is 1? No, remember this was the little subliminal problem. This was the subliminal question, where I was just looking at the exponent. So now, the exponent is approaching 1. That is the moral of this story. The exponent is approaching 1. So if I come back to the original question, I can write equals – and I see the exponent is approaching 1, so what’s the original limit? The original limit is e1, which is just e. So this limit, this thing that sort of looks like 1?, this particular one ends up approaching e. Sort of surprising to see e making a guest appearance here. There were no logs or anything in here, but we introduced those to solve the question. And, in fact, I’m going to tell you something really neat. Many people actually take whatever number that is and define that answer to be e. So, in fact, many people actually use this limit as the definition for what e is. Now you might think of e as being 2.718, blah, blah, blah. But a lot of people actually say e is actually equal to that. So this fact is a real, real important, important summit.
I’ll tell you what – why don’t we try one last one together at the next lecture. Join me, will you? I’ll see you there. Bye.

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