
“real: chapter_05” — 2011/5/22 — 22:55 — page 49 — #49
Limits and Continuity 5-49
UNSOLVED EXERCISES
1. Prove that if lim
x→0
f (x) = 0 and
lim
x→0
f (2x) − f (x)
x
= 0, then lim
x→0
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