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Calculus: Indeterminate Forms


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:53
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 95 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
Indeterminate Forms Page [1 of 2]
When we took limits in Calculus I, you know you'd sort plug in the values, see what the function's approaching and then you get an answer, sometimes. So, for example, if you just had something like the limit as x approaches 3 of x² + 1, what does that equal? Well, as x gets closer and closer to 3, x² gets closer and closer to 9 and so when you add 1 you get 10. So, this thing is approaching 10. It's not a problem, but there are some problem children in taking limits. For example, suppose when you plug in you get something awful like, ^0/[0]? Now that's not great. If it were like ^0/[7], then that limit would be just 0. If it would like ^7/[0], then that limit would be undefined or maybe infinity, it you like to think that way.
But ^0/[0] is something that we call an indeterminate form. Because what does it equal? You see, in one camp you could say, "Look, ^0/[0], I'll cancel them out and I'll get 1. So it should equal 1." Well, the other camp says, "Anything, 0 over anything is 0, so the answer should be 0." Then another camp goes, "Wait a minute, but anything over 0 is infinity." So we have three different camps and they're all fighting against each other. Right? One says it should be 1. One says it should be 0. One says it should be infinity. So, what is it? Well, it's an indeterminate form. We just don't know.
So, what do we do in that case? Well, in that case what we have to do is more work. So, let me show you an example where in fact indeterminate forms actually arise. So, let's take a look at the following, the limit as x approaches 3 of 2x² - 18 all over x - 3. Now, let's take the limit and see what we get. So, as x gets closer and closer to 3, what happens to this whole thing? Well, as x gets really, really close to 3, this term right here is getting really, really close to 9. So this whole thing is getting really, really close to 18. When I subtract this 18, the top is getting really, really close to 0. Okay, that's fine. That's fine. No need to panic quite yet.
What about the bottom? Well as x * 3, this thing, x - 3, is approaching 0. Now is the time to panic, because this is an indeterminate form. Whenever we take the limit and we see something that seems to heading towards ^0/[0] that means we don't know the answer. The limit might exist. The limit might not exist. We just don't know. So we need to do something else. Now, what did we learn in first semester Calculus? Well, one thing we learned in first semester Calculus is you can try to do some little algebraic tricks. So, here's an algebraic trick.
For example, you notice that the top is just crying out to be factored. So, let's factor the top and see what we get. If we factor the top, I could take out a 2. So, if I take out a 2 I'm left with x² -9 all divided by x - 3. Then this is the difference of two perfect squares. So, I know how to factor this really, really fast. This is the sort of speed factoring course, too--(x + 3) and (x - 3). Now this great, because in taking the limit I actually see the ^0/[0]. Do you see the ^0/[0]? It's right here. See, if x approaches 3 then I have x - 3. There's the 0 and there's the 0. So there is the ^0/[0]. That's what's giving us trouble. But look, it's a common factor. So I can now just cancel them away as long as I make a promise to everyone and the promise is that I'll never have x = 3.
But remember, the whole point of a limit is to allow us to inch up to 3 and never actually go out and touch it. We're just going to get closer and closer to it. So, if I never equal 3, I just cancel, now what's the limit? Well, I can see what the limit is. The limit would be just what this is approaching as x approaches 3. This is approaching 3 + 3, which is 6 times 2 is 12. So, this limit, in fact, equals 12. Here's an example where we first got ^0/[0] and then what happened? Well, we did a little extra work and we saw that actually the exists, it equals 12.
Now, how about this one? Here's another exotic thing that you might have seen when you were taking a look at finding asymptotes. Okay, -x³ + 9x -5 all over 10x² + 3. Now, x is approaching infinity. Now infinity is not a number, it means that x is getting arbitrarily large. Make it bigger, make it bigger, make it bigger, never stop. Don't be content with any finite number. Let it race of to the horizon and see what you get. So, what happens here? Well on the top, you see, I've got this x³. Now if x is going off and getting bigger and bigger and bigger, then x³ is just shooting out. So, this whole thing is really growing fast. This thing is just getting bigger and bigger and bigger. Now there's a negative sign there, but this thing is going off to negative infinity.
The bottom, well the bottom is actually getting really, really big too. So, what I see here is actually a limit that's seems to be heading toward infinity, divided by infinity. That is another type of an indeterminate form, because I can make the same ridiculous argument that I made before. Well, infinity over infinity--it's infinity. Infinity over infinity, if you cancel you get 1. Well, that's not right, because infinity is not a number. Infinity over anything, you might say, "That's infinity." Well, that might not be right and anything over infinity maybe that's actually 0. Well, no. When you have infinity over infinity, this is an indeterminate form. The limit might exist, it might not exist, you've got to do something else.
Well, what kind of trick can you do here? Well, one really neat trick to do is actually to notice that if I find the highest power--so I search everywhere around here for the highest power. I see an x³. That is the world's champion power, anywhere here. Look around, x³. This is an x to the first power. This is an x². There are no x's here at all. So, take x³ and divide top and bottom by that power. Now you're saying, "Divide top and bottom by that?" Well, if I multiply something by 1, it doesn't change the value. So, watch what happens if I take -x³ + 9x -5 and divide it by 10x² + 3 and then I multiply the top by ^1/[x³], then the bottom by the same thing. I've done nothing. All I've done is multiply this number by 1, something over itself. It's just 1. It doesn't change the value.
But look what happens when we distribute that ^1/[x³] throughout. When we distribute that ^1/[x³] throughout what we see is this. Well, in this term I get a complete cancellation, but don't forget that negative sign. So, here I just see -1. Here I get some cancellation. That x cancels with the x³, leaving me with an x². So, I have + ^9/[x²] and that last term ^-5/[x³]. On the bottom, when I distribute, I see a ^10/[x] and then + ^3/[x³]. So, that's what I get. Now what happens as x races off to infinity?
Well, let's take a look. This term, right here--if x is getting larger and larger and larger, then that bottom is growing. If the bottom is growing, the whole thing is getting smaller and smaller. This is like ^2/[10], ^2/[1,000,000], 2 over gazillion. That's actually approaching 0. So this term is going to 0 and this term is also going to 0 and this term is also going to 0 and this term is also going to 0. Looks like everything has been circled, but no it hasn't. The -1 remains and lives on to tell the children about the massacre. Now, what's going on here? Well, on the top I'm approaching, therefore, -1. On the bottom, I'm approaching 0. This is not an indeterminate form, because this is not ^0/[0] or infinity over infinity. This is -1 and something on the bottom that's approaching 0. That is negative infinity. So, in fact, this limit is equal to negative infinity.
So, there are examples when you get indeterminate forms when you just don't know what the answer is. Now, the question that I want to raise right now is, what if you had more exotic functions? What if I had sines and cosines or the exponential function, e^x? How could I figure out limits that remain indeterminate when I try to plug in the values? Well, it turns out that there's a method, a method that allows us to actually figure out the answers to these hard limit questions that actually just involves a little bit of differentiation. It's a really cool trick and I'll show that to you next. I'll see you there.

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