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Vojtěch Jarník IMC
2014 VJIMC
Problem 1
limit(∞) of f(x)+f'(x)/x given, find lim(∞) f(x)
limit(∞) of f(x)+f'(x)/x given, find lim(∞) f(x)
Source: VJIMC 2014 2.1
May 21, 2021
limit
limits
real analysis
Problem Statement
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb R
f
:
(
0
,
∞
)
→
R
be a differentiable function. Assume that
lim
x
→
∞
(
f
(
x
)
+
f
′
(
x
)
x
)
=
0.
\lim_{x\to\infty}\left(f(x)+\frac{f'(x)}x\right)=0.
x
→
∞
lim
(
f
(
x
)
+
x
f
′
(
x
)
)
=
0.
Prove that
lim
x
→
∞
f
(
x
)
=
0.
\lim_{x\to\infty}f(x)=0.
x
→
∞
lim
f
(
x
)
=
0.
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