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Vojtěch Jarník IMC
2013 VJIMC
Problem 1
f(x)f'(x)≥cos, f(∞)=undef. if f is bounded
f(x)f'(x)≥cos, f(∞)=undef. if f is bounded
Source: VJIMC 2013 1.1
May 30, 2021
function
calculus
limits
Problem Statement
Let
f
:
[
0
,
∞
)
→
R
f:[0,\infty)\to\mathbb R
f
:
[
0
,
∞
)
→
R
be a differentiable function with
∣
f
(
x
)
∣
≤
M
|f(x)|\le M
∣
f
(
x
)
∣
≤
M
and
f
(
x
)
f
′
(
x
)
≥
cos
x
f(x)f'(x)\ge\cos x
f
(
x
)
f
′
(
x
)
≥
cos
x
for
x
∈
[
0
,
∞
)
x\in[0,\infty)
x
∈
[
0
,
∞
)
, where
M
>
0
M>0
M
>
0
. Prove that
f
(
x
)
f(x)
f
(
x
)
does not have a limit as
x
→
∞
x\to\infty
x
→
∞
.
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