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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_05” 2011/5/22 22:55 page 49 #49
Limits and Continuity 5-49
UNSOLVED EXERCISES
1. Prove that if lim
x0
f (x) = 0 and
lim
x0
f (2x) f (x)
x
= 0, then lim
x0
f (x)
x
= 0.
2. Suppose that f is defined on (a, ) and is bounded on each finite
interval (a, b), a < b.If lim
x→∞
(f (x + 1) f (x)) = l, then show that
lim
x→∞
f (x)
x
= l.
3. Suppose that f is defined on (a, ) and is bounded above on each finite
interval (a, b), a < b.If lim
x→∞
(f (x +1) f (x)) =−∞, then show that
lim
x→∞
f (x)
x
=−∞.
4. Suppose that f is defined on (a, ) and is bounded below on each finite
interval (a, b), a < b.If lim
x→∞
(f (x + 1) f (x)) =∞, then show that
lim
x→∞
f
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Publisher Resources

ISBN: 9781299447561Publisher Website