5128_Ch08_434-471

448 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
) .
EXAMPLE 7
Find lim
x→1 ( 1 1
ln x x 1
Working with Indeterminate Form
SOLUTION
Combining the two fractions converts the indeterminate form to 00, to which
we can apply l’Hôpital’s Rule.
lim
x→1 ( 1 1
ln x x
) 1
lim x 1 ln
x
Now 0 x→1 x 1 ln
x
0
1 1x
lim
x→1
x 1
ln x
x
x 1
lim Still 0 x→1 x ln x x 1 0
lim
x→1 2
1ln x
1 2 Now try Exercise 19.
Indeterminate Forms 1 , 0 0 , 0
Limits that lead to the indeterminate forms 1 ,0 0 , and 0 can sometimes be handled by
taking logarithms first. We use l’Hôpital’s Rule to find the limit of the logarithm and then
exponentiate to reveal the original function’s behavior.
Since b e ln b for every positive number
b, we can write fx as
fx eln fx
for any positive function fx.
lim ln f x L ⇒ lim f x lim e ln f x e L
x→a x→a x→a
Here a can be finite or infinite.
In Section 1.3 we used graphs and tables to investigate the values of f x 1 1x x
as x→. Now we find this limit with l’Hôpital’s Rule.
EXAMPLE 8
Find
lim
x→ (
Working with Indeterminate Form 1
x
1 1 x ) .
SOLUTION
Let f x 1 1x x . Then taking logarithms of both sides converts the indeterminate
form 1 to 00, to which we can apply l’Hôpital’s Rule.
x ln
ln f x ln ( 1 1 x ) x ln ( 1 1 x )
( 1 1
x )
1 x
continued