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Calculus: Basic Uses of L'Hopital's Rule

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About this Lesson

  • Type: Video Tutorial
  • Length: 10:54
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 116 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: Indeterminate Quotients (4 lessons, $6.93)

In this lesson, we will look at applications of L'Hopital's rule, when to use it, what to watch out for, etc. When assessing limits, start by plugging in using substitution to evaluate them. If this gives you an indeterminate form, apply L'Hôpital's rule. The L'Hôpital 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). Note that the rule may mislead you if you misapply it to a limit that doesn't produce 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.

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L’Hospital’s Rule
Indeterminate Quotients
Basic Uses of L’Hospital’s Rule Page [1 of 2]
So let’s take a look at some really sort of basic examples before we get really revved up to look at some more interesting ones, where we can actually apply L'Hospital's rule because we have an indeterminate form. So, let’s see, let’s consider the following, limit as x approaches -2 of x² + 3x + 2 all divided by x² + x - 2. Now always in life when you’re faced with a limit the first thing you should do is see if the limit exists just by letting x inch up to -2. So, let’s do that right now. If we let x inch up to -2, what do I see here? Well this is inching up to -2², which is 4. So, I’ve got a 4 and that 2 makes a 6. Then this is inching up to -6 and so I see actually 0 on the top, 4 - 6 + 2 is approaching 0. On the bottom I’m seeing a 4 - 2 - 2, that’s 4 - 4 is 0. So I see an indeterminate form of the form 0/0. What to do, what to do?
Well, let’s apply L'Hospital's rule. Now, how does this work? All I do is formally take the derivative of the top, formally take the derivative of the bottom and try the limit again. So, let’s try it. This equals the limit as x approaches -2 and now I’m going to write the derivative of this, which is 2x + 3 and the derivative of the bottom is 2x + 1. Now, what’s the limit now? Well, let’s let x approach -2. If I let x approach -2, on the top I see -4 + 3. So that’s actually -1. So, hold the press, stop everything. We’re not going to have another indeterminate form. Whether this limit exists or not, it’s not going to be indeterminate, because right now I’m going to be able to say that the top of this is approaching -1. Well, what about the bottom? Well, as x approaches -2 this is approaching -4 + 1 would be about -3. So that limit actually exists and equals 1/3. So there’s the answer.
Okay, let’s try another one. How about this one? How about the limit as x approaches infinity of x103 - 1,000--oh this is a big, divided by 103x + 2x103? Now I classify this as a mean problem, because first of all there are big numbers. Why do we have big numbers? Why can’t we just have 1’s and 2’s? Okay, that’s the first question. Second of all, the thing that’s really sneaky about this problem is that the highest power of x appearing in the denominator is not up front where you usually expect it. It was placed way back here. So, that’s a little bit sneaky. Let’s see if we can make a guess as to what we think this answer is. We’re going to let x race off to infinity. If I let x get really, really big this thing is just going off to infinity really, really fast and the bottom is also going to infinity. So this is an indeterminate form of the variety infinity over infinity. So, it needs more work and I want to apply L'Hospital's rule.
Before I do that I actually want to tell you how you can make a good guess as what the answer may be. Let’s look at this and reason together. Look, how is this thing growing? This thing’s basically growing like this. This -1,000 is almost contributing nothing. If I’m going off to infinity and I’m taking a huge number and raising it to the 103 power--that's big. Are you going to notice a -1,000? You wouldn’t even see it. That’s not even a piece of sand on the Sahara Desert compared to what this is going on. This is what’s going on. That’s the whole thing. What about the bottom? Well, even though this is going to infinity, this is going to infinity at such a greater rate. It’s going to infinity so fast that even this is negligible. So, really all I see when I look at this, personally, I’m letting you in on a little secret, all I see is that. If you notice now I can sort of cancel away those terms. What am I left with? I’m left with just ½. So, I’m going to guess the answer is going to be ½. Because the rates of growth on top and bottom are the same and so all I do is look at the coefficients.
Let’s verify that guess or see if I’m wrong by using L'Hospital's rule. So it’s always good to sort of think about what the answer might be to see if your answer is reasonable. So, let’s take the derivative of the top, derivative of the bottom. The derivative of the top, that’s pretty easy. It’s 103x102, which is by the way the temperature here right now, 102. Then the derivative of the bottom is 103 + and then 2 times 103, which is about 206x and again to the temperature of the room. Okay, let’s now take the limit again and let x race off to infinity. I still see something growing arbitrarily large divided by something that’s growing arbitrarily large. This is still an indeterminate form of the variety infinity over infinity. What to do, what to do?
Well, let’s use L'Hospital's rule once again. Well, if you take the limit as x approaches infinity, now the top I’m going to have 103 x 102. Well, what’s that? Well that’s about 10,506--that is a big number, x101 divided by--well, this is derivative, this is 0 and I have 206 multiplied by 102. What’s that? Well, that, well just double this. So 21,012x101. Now notice I can cancel these things away. If I cancel those away, I'm just left with 10,506 divided by 21,012. So that’s the limit and that might not look particularly nice, but then if you cancel away the common factor of 10,506 you see ½. So our guess was actually pretty good. Our guess was good. That will always work, by the way. So, that’s a great way to check if your answer is okay. So there you go.
Okay, let’s try a couple of more really fast. Let’s try the limit as x approaches negative infinity. I’m going off to the left-hand horizon. Let’s take a look at 2x3 - 4 divided by 1 million. Why do I do this to myself? You know how many 0’s you need for a million? Okay, that many 0’s. x2 + 3. Now, plainly the top is going off to some infinity. Now actually what infinity is it going off to? If you’re taking a number that’s getting really, really big, but there’s a negative sign in front of it, when I cube it it’ll still have that negative sign in front. So this is going to negative infinity, but the bottom, since I’m squaring, I get positive numbers. It’s going to positive infinity. So this is like negative infinity over positive infinity, that is an indeterminate form. To our rescue comes L'Hospital's rule.
So, let’s take the derivative of the top, derivative of the bottom and see what happens. The derivative of the top is going to be 6x2. The derivative of the bottom is 2 million, that’s a big number, x. Now, I can cancel away and then just one of these is left. So, I’m left with 6x over 2 million. Well, I know this is not going to be an indeterminate form because the bottom is 2 million. It’s not infinity. It’s not 0. So this limit--well, we’re going to determine what the answer is right now. What’s happening? As x goes to negative infinity, the top is going to negative infinity. Where it’s like negative infinity times 6. That’s negative infinity, but the bottom is just staying at 2 million. So, this answer actually, you could say, does not exist or equals negative infinity.
Moral of the story here, even though you have whooping coefficient all that really matters is the exponent and this thing is a bigger exponent, so that should dominate. Since the thing is going to negative infinity, the whole thing pulls down to negative infinity. So, a little moral, life lesson there. Okay, let’s try one last one real fast. Limit as x approaches infinity of . Now, I’m actually going to do this in character. So, I'm going to take on the persona of a calculus student. Not you, but someone else. Here I go. In fact, I’m going to use the calculus student nom de plume, which means I’m going to change colors of my pen.
Take the derivative of the top and the bottom. So, the derivative of the top is 0 and the derivative of the bottom is 2x. I can do that again and I see 0/2. So, maybe I can get 0 now. Okay. So, let’s take a look at the work of the student. Well, the work of the student was not great, because in this first step, what was the point of this? Well, the point of this was to use L'Hospital's rule. But to use L'Hospital's rule we have to make sure we have an indeterminate form. Is this really an indeterminate form? The answer is no, because I’m not approaching 0/0. I'm not approaching infinity over infinity or negative infinity over infinity of negative infinity over negative infinity or infinity over negative infinity. So that is actually not appropriate. L'Hospital's rule is not appropriate. This limit should be done just immediately.
If you look at this, what happens? If the top is approaching 15 and the bottom is getting arbitrarily large, then this whole thing is indeed going to 0. So this person actually got the right answer, but for the absolute wrong reasons. So you have to be really, really careful. Sometimes this thing will work. Sometimes it won’t. Do you want to risk it? No. So make sure you always start off with an indeterminate form before you actually embark upon taking derivatives of tops and bottoms. Okay, up next I want to take a look at some real classic and exotic examples where L'Hospital's rule will come in handy. So far everything we’ve looked at have been just polynomials divided by polynomials and could have done it the old fashion way. Now we’re going to take a look at really cool functions.

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