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The Chinese Mathematics Competition
2021 The Chinese Mathematics Competition
2021 The Chinese Mathematics Competition
Part of
The Chinese Mathematics Competition
Subcontests
(11)
3
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2021 CMC (Non-math Major) Mark-up Test
Part 1: Fill in the blanks. 1. Let
x
0
=
1
x_0=1
x
0
=
1
,
x
n
=
l
n
(
1
+
x
n
−
1
)
x_n=ln(1+x_{n-1})
x
n
=
l
n
(
1
+
x
n
−
1
)
(
n
≥
1
n\geq1
n
≥
1
). Find
lim
n
→
+
∞
n
x
n
\lim_{n \to +\infty}nx_n
lim
n
→
+
∞
n
x
n
.2. Find
I
=
∫
0
π
2
c
o
s
x
1
+
t
a
n
x
d
x
I=\int_{0}^{\frac{\pi}{2}}\frac{cos x}{1+tan x}dx
I
=
∫
0
2
π
1
+
t
an
x
cos
x
d
x
.3. Let
L
:
{
2
x
−
4
y
+
z
=
0
3
x
−
y
−
2
z
=
9
L:\begin{cases}2x-4y+z=0\\3x-y-2z=9\end{cases}
L
:
{
2
x
−
4
y
+
z
=
0
3
x
−
y
−
2
z
=
9
and plane
π
:
4
x
−
y
+
z
=
1
\pi:4x-y+z=1
π
:
4
x
−
y
+
z
=
1
. Find the equation of projection of straight line
L
L
L
on the plane
π
\pi
π
.4. Find
∑
n
=
1
+
∞
a
r
c
t
a
n
2
4
n
2
+
4
n
+
1
.
\sum_{n=1}^{+\infty}arctan\frac{2}{4n^2+4n+1}.
∑
n
=
1
+
∞
a
rc
t
an
4
n
2
+
4
n
+
1
2
.
5. Solve
(
x
+
1
)
d
y
d
x
+
1
=
2
e
−
y
(x+1)\frac{dy}{dx}+1=2e^{-y}
(
x
+
1
)
d
x
d
y
+
1
=
2
e
−
y
where
y
(
0
)
=
0.
y(0)=0.
y
(
0
)
=
0.
Part 2: Proof-based Questions 6. Let
f
(
x
)
=
−
1
2
(
1
+
1
e
)
+
∫
−
1
1
∣
x
−
t
∣
e
−
t
2
d
t
f(x)=-\frac{1}{2}(1+\frac{1}{e})+\int_{-1}^{1}|x-t|e^{-t^2}dt
f
(
x
)
=
−
2
1
(
1
+
e
1
)
+
∫
−
1
1
∣
x
−
t
∣
e
−
t
2
d
t
. Prove that
f
(
x
)
f(x)
f
(
x
)
has only two real roots in the interval
(
−
1
,
1
)
(-1, 1)
(
−
1
,
1
)
.7. Let
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
be a function that exists a continuous second-order partial differentiation in closed region
D
=
{
(
x
,
y
)
∣
x
2
+
y
2
≤
1
}
D=\{(x,y)|x^2+y^2\leq1\}
D
=
{(
x
,
y
)
∣
x
2
+
y
2
≤
1
}
such that
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
=
x
2
+
y
2
\frac{\partial^2f }{\partial x^2}+\frac{\partial^2f }{\partial y^2}=x^2+y^2
∂
x
2
∂
2
f
+
∂
y
2
∂
2
f
=
x
2
+
y
2
. Find
lim
r
→
0
+
∫
∫
x
2
+
y
2
≤
r
2
(
x
∂
f
∂
x
+
y
∂
f
∂
y
)
d
x
d
y
(
t
a
n
r
−
s
i
n
r
)
2
\lim_{r \to 0^+} \frac{\int\int_{x^2+y^2\leq r^2}^{}(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y})dxdy}{(tan r-sin r)^2}
lim
r
→
0
+
(
t
an
r
−
s
in
r
)
2
∫
∫
x
2
+
y
2
≤
r
2
(
x
∂
x
∂
f
+
y
∂
y
∂
f
)
d
x
d
y
.8. It is given that every orientable smooth closed surface
S
S
S
in half space in
R
3
R^3
R
3
{
(
x
,
y
,
z
)
∈
R
3
∣
x
>
0
}
\{ (x,y,z)\in R^3 |x>0\}
{(
x
,
y
,
z
)
∈
R
3
∣
x
>
0
}
exists
∫
∫
S
x
f
′
(
x
)
d
y
d
z
+
y
(
x
f
(
x
)
−
f
′
(
x
)
)
d
z
d
x
−
x
z
(
s
i
n
x
+
f
′
(
x
)
)
d
x
d
y
=
0
\int\int_{S}^{}xf'(x)dydz+y(xf(x)-f'(x))dzdx-xz(sin x+f'(x))dxdy=0
∫
∫
S
x
f
′
(
x
)
d
y
d
z
+
y
(
x
f
(
x
)
−
f
′
(
x
))
d
z
d
x
−
x
z
(
s
in
x
+
f
′
(
x
))
d
x
d
y
=
0
, where
f
f
f
is twice continuously differentiable on the interval
(
0
,
+
∞
)
(0,+\infty)
(
0
,
+
∞
)
and
lim
x
→
0
+
f
(
x
)
=
lim
x
→
0
+
f
′
(
x
)
=
0
\lim_{x \to 0^+} f(x)=\lim_{x \to 0^+} f'(x)=0
lim
x
→
0
+
f
(
x
)
=
lim
x
→
0
+
f
′
(
x
)
=
0
. Find
f
(
x
)
f(x)
f
(
x
)
.9. Let
f
(
x
)
=
∫
0
x
(
1
−
⌊
u
⌋
u
)
d
u
f(x)=\int_{0}^{x}(1-\frac{\left\lfloor u \right\rfloor}{u})du
f
(
x
)
=
∫
0
x
(
1
−
u
⌊
u
⌋
)
d
u
. Discuss the convergence and divergence of
∫
1
+
∞
e
f
(
x
)
x
p
c
o
s
(
x
2
−
1
x
2
)
d
x
\int_{1}^{+\infty}\frac{e^{f(x)}}{x^p}cos(x^2-\frac{1}{x^2})dx
∫
1
+
∞
x
p
e
f
(
x
)
cos
(
x
2
−
x
2
1
)
d
x
, where
p
p
p
is positive number.10. Let
a
n
{a_n}
a
n
be a positive sequence that is monotonically decreasing and tends to
0
0
0
and
f
(
x
)
=
∑
n
=
1
∞
a
n
n
x
n
f(x)=\sum_{n=1}^{\infty}a_n^nx^n
f
(
x
)
=
∑
n
=
1
∞
a
n
n
x
n
. Prove that if the series
∑
n
=
1
∞
a
n
\sum_{n=1}^{\infty}a_n
∑
n
=
1
∞
a
n
diverges, then integral
∫
1
+
∞
l
n
f
(
x
)
x
2
d
x
\int_{1}^{+\infty}\frac{lnf(x)}{x^2}dx
∫
1
+
∞
x
2
l
n
f
(
x
)
d
x
diverges too.
2021 CMC (Math Major) Set A
1.
1.
1.
It is given that at least one of
a
,
b
,
c
∈
R
a,b,c\in \mathbb{R}
a
,
b
,
c
∈
R
is nonnegative. Find the equation of surface of rotation when the straight line
x
−
1
a
=
y
−
1
b
=
z
−
1
c
\frac{x-1}{a}=\frac{y-1}{b}=\frac{z-1}{c}
a
x
−
1
=
b
y
−
1
=
c
z
−
1
rotates about the
z
z
z
-axis.
2.
2.
2.
Let
B
⊂
R
n
(
n
≥
2
)
B\subset R^n(n\ge 2)
B
⊂
R
n
(
n
≥
2
)
be a unit open ball and function
u
,
v
u,v
u
,
v
is continuous on
b
‾
\overline{b}
b
and twice continuously differentiable in
B
B
B
. It satisfies the condition (i)
−
Δ
u
−
(
1
−
u
2
−
v
2
)
u
=
0
-\Delta u-(1-u^2-v^2)u=0
−
Δ
u
−
(
1
−
u
2
−
v
2
)
u
=
0
(ii)
−
Δ
v
−
(
1
−
u
2
−
v
2
)
v
=
0
-\Delta v-(1-u^2-v^2)v=0
−
Δ
v
−
(
1
−
u
2
−
v
2
)
v
=
0
(iii)
u
(
x
)
=
v
(
x
)
=
0
u(x)=v(x)=0
u
(
x
)
=
v
(
x
)
=
0
, where
x
∈
∂
B
x\in \partial B
x
∈
∂
B
. Also,
x
=
(
x
1
,
x
2
,
⋯
,
x
n
)
x=(x_1,x_2,\cdots,x_n)
x
=
(
x
1
,
x
2
,
⋯
,
x
n
)
,
Δ
u
=
∂
2
u
∂
x
1
2
+
∂
2
u
∂
x
2
2
+
⋯
+
∂
2
u
∂
x
n
2
\Delta u=\frac{\partial ^2u}{\partial x^2_1}+\frac{\partial ^2u}{\partial x^2_2}+\cdots+\frac{\partial ^2u}{\partial x^2_n}
Δ
u
=
∂
x
1
2
∂
2
u
+
∂
x
2
2
∂
2
u
+
⋯
+
∂
x
n
2
∂
2
u
. (
∂
B
\partial B
∂
B
is the boundary of
B
B
B
.) Prove that
u
2
(
x
)
+
v
2
(
x
)
≤
1
u^2(x)+v^2(x)\leq1
u
2
(
x
)
+
v
2
(
x
)
≤
1
(
∀
x
∈
B
‾
\forall x\in \overline{B}
∀
x
∈
B
).
3.
3.
3.
Let
f
(
x
)
=
x
2
021
+
a
2020
x
2020
+
a
2019
x
2019
+
⋯
+
a
2
x
2
+
a
1
x
+
a
0
f(x)=x^2021+a_{2020}x^{2020}+a_{2019}x^{2019}+\cdots+a_2x^2+a_1x+a_0
f
(
x
)
=
x
2
021
+
a
2020
x
2020
+
a
2019
x
2019
+
⋯
+
a
2
x
2
+
a
1
x
+
a
0
be a polynomial with integer coefficients and
a
0
≠
0
a_0\neq 0
a
0
=
0
. For any
0
≤
k
≤
2020
0 \leq k \leq 2020
0
≤
k
≤
2020
, there exists
∣
a
k
∣
≤
40
|a_k|\leq 40
∣
a
k
∣
≤
40
. Prove that it is not possible for all roots of
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
to be a real number.
4.
4.
4.
Let
P
P
P
be an unitary and symmetric matrix. Prove that there exists an inverse complex matrix
Q
Q
Q
such that
P
=
Q
‾
Q
−
1
P=\overline{Q}Q^{-1}
P
=
Q
Q
−
1
.
5.
5.
5.
Let
α
\alpha
α
be an integer greater than
1
1
1
. Prove that (1)
∫
0
+
∞
d
x
∫
0
+
∞
e
−
t
α
x
s
i
n
x
d
t
=
∫
0
+
∞
d
t
∫
0
+
∞
e
−
t
α
x
s
i
n
x
d
x
\int_{0}^{+\infty}dx\int_{0}^{+\infty}e^{{-t}^{\alpha}x}sinxdt=\int_{0}^{+\infty}dt\int_{0}^{+\infty}e^{{-t}^{\alpha}x}sinxdx
∫
0
+
∞
d
x
∫
0
+
∞
e
−
t
α
x
s
in
x
d
t
=
∫
0
+
∞
d
t
∫
0
+
∞
e
−
t
α
x
s
in
x
d
x
. (2) Evaluate
∫
0
+
∞
s
i
n
x
3
d
x
⋅
∫
0
+
∞
s
i
n
x
3
2
d
x
\int_{0}^{+\infty}sin x^3dx\cdot \int_{0}^{+\infty}sin x^{\frac{3}{2}}dx
∫
0
+
∞
s
in
x
3
d
x
⋅
∫
0
+
∞
s
in
x
2
3
d
x
.
6.
6.
6.
It satisfies:
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
,
f
′
(
x
)
≥
6
+
f
(
x
)
−
f
2
(
x
)
f'(x)\ge 6+f(x)-f^2(x)
f
′
(
x
)
≥
6
+
f
(
x
)
−
f
2
(
x
)
and
g
′
(
x
)
≤
6
+
g
(
x
)
−
g
2
(
x
)
g'(x)\le 6+g(x)-g^2(x)
g
′
(
x
)
≤
6
+
g
(
x
)
−
g
2
(
x
)
. Prove that (1)
∀
x
∈
(
0
,
1
)
\forall x\in (0, 1)
∀
x
∈
(
0
,
1
)
and
x
∈
R
x\in \mathbb{R}
x
∈
R
, there exists
ξ
∈
(
−
∞
,
x
)
\xi\in (-\infty,x)
ξ
∈
(
−
∞
,
x
)
such that
f
(
ξ
)
≥
3
−
ε
f(\xi)\ge 3-\varepsilon
f
(
ξ
)
≥
3
−
ε
. (2)
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
,
f
(
x
)
≥
3
f(x) \ge 3
f
(
x
)
≥
3
. (3)
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
, there exists
η
∈
(
−
∞
,
x
)
\eta\in (-\infty,x)
η
∈
(
−
∞
,
x
)
such that
g
(
η
)
≤
3
g(\eta)\le 3
g
(
η
)
≤
3
. (4)
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
,
g
(
x
)
≤
3
g(x) \le 3
g
(
x
)
≤
3
.
2021 CMC (Math Major) Set B
1. Let sphere
S
:
x
2
+
y
2
+
z
2
=
1
S:x^2+y^2+z^2=1
S
:
x
2
+
y
2
+
z
2
=
1
. Find the equation of pyramid with vertex
M
0
(
0
,
0
,
a
)
M_0(0,0,a)
M
0
(
0
,
0
,
a
)
(
a
∈
R
a\in\mathbb{R}
a
∈
R
,
∣
a
∣
>
1
|a|>1
∣
a
∣
>
1
) and cut
S
S
S
. 2. Same as 2021 CMC (Math Major) Set A Problem 2. 3. Same as 2021 CMC (Math Major) Set A Problem 3. 4. Let
R
=
0
,
1
,
−
1
R={0,1,-1}
R
=
0
,
1
,
−
1
and
S
=
{
d
e
t
(
a
i
j
)
3
×
3
∣
a
i
j
∈
R
}
S=\{ det(a_{ij})_{3\times 3}|a_{ij}\in R\}
S
=
{
d
e
t
(
a
ij
)
3
×
3
∣
a
ij
∈
R
}
. Prove that
S
=
−
4
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
4
S={-4,-3,-2,-1,0,1,2,3,4}
S
=
−
4
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
4
. 5. Let function
f
f
f
is defined in
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
and
f
f
f
is continuously differentiable in some domain of
x
=
0
x=0
x
=
0
. Also,
lim
x
→
0
f
(
x
)
x
=
a
>
0
\lim_{x \to 0} \frac{f(x)}{x}=a>0
lim
x
→
0
x
f
(
x
)
=
a
>
0
. Prove that if series
∑
n
=
1
∞
(
−
1
)
n
f
(
1
n
)
\sum_{n=1}^{\infty}(-1)^nf(\frac{1}{n})
∑
n
=
1
∞
(
−
1
)
n
f
(
n
1
)
converges, then
∑
n
=
1
∞
f
(
1
n
)
\sum_{n=1}^{\infty}f(\frac{1}{n})
∑
n
=
1
∞
f
(
n
1
)
diverges. 6. Let
f
(
x
)
=
l
n
∑
n
=
1
∞
e
n
x
n
2
f(x)=ln\sum_{n=1}^{\infty}\frac{e^{nx}}{n^2}
f
(
x
)
=
l
n
∑
n
=
1
∞
n
2
e
n
x
. Prove that function
f
f
f
is strictly convex in
(
−
∞
,
0
)
(- \infty ,0)
(
−
∞
,
0
)
and for any
ξ
∈
(
−
∞
,
0
)
\xi\in(-\infty ,0)
ξ
∈
(
−
∞
,
0
)
, there exists
x
1
x_1
x
1
,
x
2
∈
(
−
∞
,
0
)
x_2\in(-\infty,0)
x
2
∈
(
−
∞
,
0
)
such that
f
′
(
ξ
)
=
f
(
x
2
)
−
f
(
x
1
)
x
2
−
x
1
f'(\xi)=\frac{f(x_2)-f(x_1)}{x_2-x_1}
f
′
(
ξ
)
=
x
2
−
x
1
f
(
x
2
)
−
f
(
x
1
)
.
Problem 8
1
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2021 CMC (Non-Math Major) Problem 8
Consider a homogeneous function with degree
4
4
4
.
f
(
x
,
y
,
z
)
=
a
1
x
4
+
a
2
y
4
+
a
3
z
4
+
3
a
4
x
2
y
2
+
3
a
5
y
2
z
2
+
3
a
6
x
2
z
2
f(x,y,z)=a_1x^4+a_2y^4+a_3z^4+3a_4x^2y^2+3a_5y^2z^2+3a_6x^2z^2
f
(
x
,
y
,
z
)
=
a
1
x
4
+
a
2
y
4
+
a
3
z
4
+
3
a
4
x
2
y
2
+
3
a
5
y
2
z
2
+
3
a
6
x
2
z
2
. Find
∯
∑
f
(
x
,
y
,
z
)
d
S
\oiint_{\sum} f(x,y,z)dS
∬
∑
f
(
x
,
y
,
z
)
d
S
, where
∑
:
x
2
+
y
2
+
z
2
=
1
\sum: x^2+y^2+z^2=1
∑
:
x
2
+
y
2
+
z
2
=
1
.
Problem 10
1
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2021 CMC (Non-Math Major) Problem 10
Let
a
n
{a_n}
a
n
and
b
n
{b_n}
b
n
be positive real sequence that satisfy the following condition: (i)
a
1
=
b
1
=
1
a_1=b_1=1
a
1
=
b
1
=
1
(ii)
b
n
=
a
n
b
n
−
1
−
2
b_n=a_n b_{n-1}-2
b
n
=
a
n
b
n
−
1
−
2
(iii)
n
n
n
is an integer larger than
1
1
1
. Let
b
n
{b_n}
b
n
be a bounded sequence. Prove that
∑
n
=
1
∞
1
a
1
a
2
⋯
a
n
\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}
∑
n
=
1
∞
a
1
a
2
⋯
a
n
1
converges. Find the value of the sum.
Problem 9
1
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2021 CMC (Non-Math Major) Problem 9
Let
f
(
x
)
f(x)
f
(
x
)
be a twice continuously differentiable function on closed interval
[
a
,
b
]
[a,b]
[
a
,
b
]
Prove that
lim
n
→
∞
n
2
[
∫
a
b
f
(
x
)
d
x
−
b
−
a
n
∑
k
=
1
n
f
(
a
+
2
k
−
1
2
n
(
b
−
a
)
)
]
=
(
b
−
a
)
2
24
[
f
′
(
b
)
−
f
′
(
a
)
]
\lim_{n \to \infty} n^2[\int_{a}^{b}f(x)dx-\frac{b-a}{n}\sum_{k=1}^{n}f(a+\frac{2k-1}{2n}(b-a))]=\frac{(b-a)^2}{24}[f'(b)-f'(a)]
lim
n
→
∞
n
2
[
∫
a
b
f
(
x
)
d
x
−
n
b
−
a
∑
k
=
1
n
f
(
a
+
2
n
2
k
−
1
(
b
−
a
))]
=
24
(
b
−
a
)
2
[
f
′
(
b
)
−
f
′
(
a
)]
Problem 7
1
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2021 CMC (Non-Math Major) Problem 7
Let
f
(
x
)
f(x)
f
(
x
)
be a bounded continuous function on
[
0
,
+
∞
)
[0,+\infty)
[
0
,
+
∞
)
. Prove that every solutions of the equation
y
′
′
+
14
y
′
+
13
y
=
f
(
x
)
y''+14y'+13y=f(x)
y
′′
+
14
y
′
+
13
y
=
f
(
x
)
are bounded continuous functions on
[
0
,
+
∞
)
[0,+\infty)
[
0
,
+
∞
)
Problem 6
1
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2021 CMC (Non-Math Major) Problem 6
Let
x
1
=
2021
x_1=2021
x
1
=
2021
,
x
n
2
−
2
(
x
n
+
1
)
x
n
+
1
+
2021
=
0
x_n^2-2(x_n+1)x_{n+1}+2021=0
x
n
2
−
2
(
x
n
+
1
)
x
n
+
1
+
2021
=
0
(
n
≥
1
n\geq1
n
≥
1
). Prove that the sequence
x
n
{x_n}
x
n
converges. Find the limit
lim
n
→
∞
x
n
\lim_{n \to \infty} x_n
lim
n
→
∞
x
n
.
Problem 4
1
Hide problems
2021 CMC Problem 4
Find the equation of cylinder that passes three straight lines
L
1
=
{
x
=
0
y
−
z
=
2
,
L
2
=
{
x
=
0
x
+
y
−
z
+
2
=
0
,
L
3
=
{
x
=
2
y
−
z
=
0
L_1= \begin{cases} x=0\\ y-z=2 \end{cases}, L_2= \begin{cases} x=0\\ x+y-z+2=0 \end{cases}, L_3= \begin{cases} x=\sqrt{2}\\ y-z=0 \end{cases}
L
1
=
{
x
=
0
y
−
z
=
2
,
L
2
=
{
x
=
0
x
+
y
−
z
+
2
=
0
,
L
3
=
{
x
=
2
y
−
z
=
0
.
Problem 5
1
Hide problems
2021 CMC (Non-Math Major) Problem 5
Let
D
=
{
(
x
,
y
)
∣
x
2
+
y
2
≤
π
}
D=\{ (x,y)|x^2+y^2\le \pi \}
D
=
{(
x
,
y
)
∣
x
2
+
y
2
≤
π
}
. Find
∬
D
(
s
i
n
x
2
c
o
s
x
2
+
x
x
2
+
y
2
)
d
x
d
y
\iint\limits_D(sin x^2cosx^2+x\sqrt{x^2+y^2})dxdy
D
∬
(
s
in
x
2
cos
x
2
+
x
x
2
+
y
2
)
d
x
d
y
.
Problem 3
1
Hide problems
2021 CMC (Non-Math Major) Problem 3
Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function, where
f
(
0
)
≠
0
f(0)\neq0
f
(
0
)
=
0
. Find
lim
x
→
0
2
∫
0
x
(
x
−
t
)
f
(
t
)
d
t
x
∫
0
x
f
(
x
−
t
)
d
t
\lim_{x \to 0} \frac{2\int_{0}^{x}(x-t)f(t)dt}{x\int_{0}^{x}f(x-t)dt}
lim
x
→
0
x
∫
0
x
f
(
x
−
t
)
d
t
2
∫
0
x
(
x
−
t
)
f
(
t
)
d
t
.
Problem 2
1
Hide problems
2021 CMC (Non-Math Major) Problem 2
Let
z
=
z
(
x
,
y
)
z=z(x,y)
z
=
z
(
x
,
y
)
be implicit function with two variables from
2
s
i
n
(
x
+
2
y
−
3
z
)
=
x
+
2
y
−
3
z
2sin(x+2y-3z)=x+2y-3z
2
s
in
(
x
+
2
y
−
3
z
)
=
x
+
2
y
−
3
z
. Find
∂
z
∂
x
+
∂
z
∂
y
\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}
∂
x
∂
z
+
∂
y
∂
z
.
Problem 1
1
Hide problems
2021 CMC (Non-Math Major) Problem 1
Evaluate
lim
x
→
+
∞
x
2
+
x
+
1
x
−
l
n
(
e
x
+
x
)
x
\lim_{x \to +\infty}\sqrt{x^2+x+1}\frac{x-ln(e^x+x)}{x}
lim
x
→
+
∞
x
2
+
x
+
1
x
x
−
l
n
(
e
x
+
x
)
.