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Chapter 12: Who Needs Freud? Using the Integral to Solve Your Problems

Getting Your Hopes Up with L’Hôpital’s Rule

This powerful little rule enables you to easily compute limits that are either difficult or
impossible without it.

L’Hôpital’s Rule: When plugging the arrow-number into a limit expression gives you 0/0
or /3 3, you replace the numerator and denominator with their respective derivatives
and do the limit problem again — repeating this process if necessary — until you
arrive at a limit you can solve.
If you’re wondering why this limit rule is in the middle of this chapter about integra-
tion, it’s because you need L’Hôpital’s Rule for the next section and the next chapter.

Q. What’s lim x ? Q. What’s lim x e i?
x
2
-
_
x " 3 logx x " 3
A. The limit is 3. A. The limit is 0.
1. Plug 3 into x: 1. Plug 3 into x.
3
You get 3 . Not an answer, but just what You get 3 $ 0, one of the unacceptable
you want for L’Hôpital’s Rule. forms.
1
x
-
2. Replace the numerator and denominator 2. Rewrite e as e x to produce an
2
of the limit fraction with their respective acceptable form: lim x x .
derivatives. x " 3 e
Plugging in now gives you what you
1
= lim = lim x ln10h 3
^
x " 3 1 x " 3 need, 3 .
x ln10
3. Replace numerator and denominator
3. Now you can plug in. with their derivatives.
=
= 3 $ ln10 3 2 x
= lim x
x " 3 e
Remember: If substituting the
arrow-number into x gives you 4. Plugging in gives you 3 again, so you
3
!3 $ , 0 3 - 3 , 1 !3 , 0 0 , or ! 3 — the use L’Hôpital’s Rule a second time.
0
so-called unacceptable forms — instead
0 !3 = lim 2 x = 2 = 2 = 0
of one of the acceptable forms, or , e e 3 3
0 !3 x " 3
you have to manipulate the limit problem
to convert it into one of the acceptable
forms.