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IMC
1994 IMC
5
IMC 1994 D1 P5
IMC 1994 D1 P5
Source:
March 6, 2017
IMC
calculus
real analysis
Problem Statement
a) Let
f
∈
C
[
0
,
b
]
f\in C[0,b]
f
∈
C
[
0
,
b
]
,
g
∈
C
(
R
)
g\in C(\mathbb R)
g
∈
C
(
R
)
and let
g
g
g
be periodic with period
b
b
b
. Prove that
∫
0
b
f
(
x
)
g
(
n
x
)
d
x
\int_0^b f(x) g(nx)\,\mathrm dx
∫
0
b
f
(
x
)
g
(
n
x
)
d
x
has a limit as
n
→
∞
n\to\infty
n
→
∞
and
lim
n
→
∞
∫
0
b
f
(
x
)
g
(
n
x
)
d
x
=
1
b
∫
0
b
f
(
x
)
d
x
⋅
∫
0
b
g
(
x
)
d
x
\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx
n
→
∞
lim
∫
0
b
f
(
x
)
g
(
n
x
)
d
x
=
b
1
∫
0
b
f
(
x
)
d
x
⋅
∫
0
b
g
(
x
)
d
x
b) Find
lim
n
→
∞
∫
0
π
sin
x
1
+
3
cos
2
n
x
d
x
\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx
n
→
∞
lim
∫
0
π
1
+
3
cos
2
n
x
sin
x
d
x
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