James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A46
appendix F Proofs of Theorems
Proof of L’Hospital’s RuLe We are assuming that lim x l a f sxd − 0 and
lim x l a tsxd − 0. Let
f 9sxd
L − lim
x l a t9sxd
We must show that lim x l a f sxdytsxd − L. Define
Fsxd −H f sxd
0
if x ± a
if x − a
Gsxd −H tsxd
0
if x ± a
if x − a
Then F is continuous on I since f is continuous on hx [ I | x ± aj and
lim Fsxd − lim f sxd − 0 − Fsad
x l a x l a
Likewise, G is continuous on I. Let x [ I and x . a. Then F and G are continuous on
fa, xg and differentiable on sa, xd and G9 ± 0 there (since F9 − f 9 and G9 − t9). Therefore,
by Cauchy’s Mean Value Theorem, there is a number y such that a , y , x and
F9syd Fsxd 2 Fsad
−
G9syd Gsxd 2 Gsad − Fsxd
Gsxd
Here we have used the fact that, by definition, Fsad − 0 and Gsad − 0. Now, if we let
x l a 1 , then y l a 1 (since a , y , x), so
lim
x l a 1
f sxd
tsxd − lim
x l a 1
Fsxd
Gsxd − lim F9syd
y l a 1 G9syd − lim
y l a 1
f 9syd
t9syd
− L
A similar argument shows that the left-hand limit is also L. Therefore
lim
x l a
f sxd
tsxd − L
This proves l’Hospital’s Rule for the case where a is finite.
If a is infinite, we let t − 1yx. Then t l 0 1 as x l `, so we have
lim
x l `
f sxd
tsxd − lim
t l 0 1
f s1ytd
ts1ytd
f 9s1ytds21yt 2 d
− lim
t l 0 1
t9s1ytds21yt 2 d
f 9s1ytd
− lim
t l 0 1
t9s1ytd − lim f 9sxd
x l ` t9sxd
(by l’Hospital’s Rule for finite a)
■
Section 11.8
In order to prove Theorem 11.8.4, we first need the following results.
Theorem
1. If a power series o c n x n converges when x − b (where b ± 0), then it converges
whenever | x | , | b | .
2. If a power series o c n x n diverges when x − d (where d ± 0), then it diverges
whenever | x | . | d | .
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.