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2001 IMC
6
IMC 2001 Problem 6
IMC 2001 Problem 6
Source: IMC 2001 Day 1 Problem 6
October 30, 2020
function
real analysis
Problem Statement
Suppose that the differentiable functions
a
,
b
,
f
,
g
:
R
→
R
a, b, f, g:\mathbb{R} \rightarrow \mathbb{R}
a
,
b
,
f
,
g
:
R
→
R
satisfy
f
(
x
)
≥
0
,
f
′
(
x
)
≥
0
,
g
(
x
)
≥
0
,
g
′
(
x
)
≥
0
for all
x
∈
R
,
f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R},
f
(
x
)
≥
0
,
f
′
(
x
)
≥
0
,
g
(
x
)
≥
0
,
g
′
(
x
)
≥
0
for all
x
∈
R
,
lim
x
→
∞
a
(
x
)
=
A
≥
0
,
lim
x
→
∞
b
(
x
)
=
B
≥
0
,
lim
x
→
∞
f
(
x
)
=
lim
x
→
∞
g
(
x
)
=
∞
,
\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,
x
→
∞
lim
a
(
x
)
=
A
≥
0
,
x
→
∞
lim
b
(
x
)
=
B
≥
0
,
x
→
∞
lim
f
(
x
)
=
x
→
∞
lim
g
(
x
)
=
∞
,
and
f
′
(
x
)
g
′
(
x
)
+
a
(
x
)
f
(
x
)
g
(
x
)
=
b
(
x
)
.
\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).
g
′
(
x
)
f
′
(
x
)
+
a
(
x
)
g
(
x
)
f
(
x
)
=
b
(
x
)
.
Prove that
lim
x
→
∞
f
(
x
)
g
(
x
)
=
B
A
+
1
\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}
lim
x
→
∞
g
(
x
)
f
(
x
)
=
A
+
1
B
.
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