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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_06” 2011/5/22 23:14 page 25 #25
Differentiation 6-25
L’Hospital’s Rule cannot be applied if any of the hypothesis of
Theorem 6.5.1 is violated. In the following examples, we have limits of
the form lim
xa
f (x)
g(x )
where g
(x) = 0 frequently in the left neighbourhood
of b or lim
x0
f
(x)
g
(x)
does not exist. In these situations, Theorem 6.5.1 is
not applicable even though these limits can some times be computed
using different techniques.
Examples 6.5.3
1. lim
x→∞
xsin x
x+sin x
= lim
x→∞
f (x
g(x )
(say). Here
g
(x) = 1 + cos x = 0 for x = (2n + 1, n 1.
Hence L’Hospital’s Rule is not applicable here. However,
lim
x→∞
x sin x
x + sin x
=
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Publisher Resources

ISBN: 9781299447561Publisher Website