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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 35
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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