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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 36
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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