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2017 IMC
6
IMC 2017 Problem 6
IMC 2017 Problem 6
Source:
August 3, 2017
college contests
IMC
imc 2017
Problem Statement
Let
f
:
[
0
;
+
∞
)
→
R
f:[0;+\infty)\to \mathbb R
f
:
[
0
;
+
∞
)
→
R
be a continuous function such that
lim
x
→
+
∞
f
(
x
)
=
L
\lim\limits_{x\to +\infty} f(x)=L
x
→
+
∞
lim
f
(
x
)
=
L
exists (it may be finite or infinite). Prove that
lim
n
→
∞
∫
0
1
f
(
n
x
)
d
x
=
L
.
\lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L.
n
→
∞
lim
0
∫
1
f
(
n
x
)
d
x
=
L
.
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