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18-01F07-L36

# 18-01F07-L36 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 36 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Now, today we are continuing with this last unit. Unit 5, continued. The informal title of this unit is Dealing With Infinity. That's really the extra little piece that we're putting in to our discussions of things like limits and integrals. To start out with today, I'd like to recall for you, L'Hopital's Rule. And in keeping with the spirit here, we're just going to do the infinity / infinity case. I stated this a little differently last time, and I want to state it again today. Just to make clear what the hypotheses are and what the conclusion is. We start out with, really, three hypotheses. Two of them are kind of obvious. The three hypotheses are that f (x) tends to infinity, g ( x) tends to infinity, that's what it means to be in this infinity / infinity case. And then the last assumption is that f ' ( x) / g ' (x), tends to a limit, L. And this is all as x tends to some a. Some limit a. And then the conclusion is that f( x) / g ( x) also tends to L, as x goes to a. Now, so that's the way it is. So it's three limits. But presumably these are obvious, and this one is exactly what we were going to check anyway. Gives us this one limit. So that's the statement. And then the other little interesting point here, which is consistent with this idea of dealing with infinity, is that a equals plus or minus infinity and L equals plus or minus infinity are OK. That is, the numbers capital L, the limit capital L and the number a can also be infinite. Now in recitation yesterday, you should have discussed something about rates of growth, which follow from what I said in lecture last time and also maybe from some more detailed discussions that you had in recitation. And I'm going to introduce a notation to compare functions. Namely, we say that f ( x) is a lot less than g ( x). So this means that the limit, as it goes to infinity, this tends to 0. As x goes to infinity, this would be. So this is a notation, a new notation for us. f is a lot less than g. And it's meant to be read only asymptotically. It's only in the limit as x goes to infinity
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