18-01F07-L35 - 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 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 35 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: So we're through with techniques of integration, which is really the most technical thing that we're going to be doing. And now we're just clearing up a few loose ends about calculus. And the one we're going to talk about today will allow us to deal with infinity. And it's what's known as L'Hopital's Rule. Here's L'Hopital's Rule. And that's what we're going to do today. L'Hopital's Rule it's also known as L'Hospital's Rule. That's the same name, since the circumflex is what you put in French to omit the s. So it's the same thing, and it's still pronounced L'Hopital, even if it's got an s in it. Alright, so that's the first thing you need to know about it. And what this method does is, it's a convenient way to calculate limits including some new ones. So it'll be convenient for the old ones. There are going to be some new ones and, as an example, you can calculate x ln x as x goes to infinity. You could, whoops, that's not a very interesting one, let's try x goes to from the positive side. And you can calculate, for example, x e^ - x, as x goes to infinity. And, well, maybe I should include a few others. Maybe something like ln x / x as x goes to infinity. So these are some examples of things which, in fact, if you plug into your calculator, you can see what's happening with these. But if you want to understand them systematically, it's much better to have this tool of L'Hopital's Rule. And certainly there isn't a proof just based on a calculation in a calculator. So now here's the idea. I'll illustrate the idea first with an example. And then we'll make it systematic. And then we're going to generalize it. We'll make it much more, so when it includes these new limits, there are some little pieces of trickiness that you have to understand. So, let's just take an example that you could have done in the very first unit of this class. The limit as x goes to 1 of x ^ 10 - 1 / x ^2 - 1. So that's a limit that we could've handled. And the thing that's interesting, I mean, if you like this is in this category that we mentioned at the beginning of the course of interesting limits. What's interesting about it is that if you do this silly thing, which is
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