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Romania National Olympiad
2021 Romania National Olympiad
3
Romania National Olympiad Grade 11 P3
Romania National Olympiad Grade 11 P3
Source:
April 28, 2021
real analysis
Taylor expansion
derivative
Problem Statement
Let
f
:
R
→
R
f :\mathbb R \to\mathbb R
f
:
R
→
R
a function
n
≥
2
n \geq 2
n
≥
2
times differentiable so that:
lim
x
→
∞
f
(
x
)
=
l
∈
R
\lim_{x \to \infty} f(x) = l \in \mathbb R
lim
x
→
∞
f
(
x
)
=
l
∈
R
and
lim
x
→
∞
f
(
n
)
(
x
)
=
0
\lim_{x \to \infty} f^{(n)}(x) = 0
lim
x
→
∞
f
(
n
)
(
x
)
=
0
. Prove that:
lim
x
→
∞
f
(
k
)
(
x
)
=
0
\lim_{x \to \infty} f^{(k)}(x) = 0
lim
x
→
∞
f
(
k
)
(
x
)
=
0
for all
k
∈
{
1
,
2
,
…
,
n
−
1
}
k \in \{1, 2, \dots, n - 1\}
k
∈
{
1
,
2
,
…
,
n
−
1
}
, where
f
(
k
)
f^{(k)}
f
(
k
)
is the
k
k
k
- th derivative of
f
f
f
.
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