文章目录
第五节 极限运算法则习题 1-5
第五节 极限运算法则
定理1 两个无穷小的和是无穷小.
定理2 有界函数与无穷小的乘积是无穷小. 推论1 常数与无穷小的乘积是无穷小. 推论2 有限个无穷小的乘积是无穷小.
定理3 如果
lim
f
(
x
)
=
A
,
lim
g
(
x
)
=
B
\lim f(x)=A,\lim g(x)=B
limf(x)=A,limg(x)=B,那么 (1)
lim
[
f
(
x
)
±
g
(
x
)
]
=
lim
f
(
x
)
±
lim
g
(
x
)
=
A
±
B
\lim [f(x)\pm g(x)]=\lim f(x)\pm\lim g(x)=A\pm B
lim[f(x)±g(x)]=limf(x)±limg(x)=A±B; (2)
lim
[
f
(
x
)
⋅
g
(
x
)
]
=
lim
f
(
x
)
⋅
lim
g
(
x
)
=
A
⋅
B
\lim [f(x)\cdot g(x)]=\lim f(x)\cdot\lim g(x)=A\cdot B
lim[f(x)⋅g(x)]=limf(x)⋅limg(x)=A⋅B; (3) 若又有
B
≠
0
B\ne0
B=0,则
lim
f
(
x
)
g
(
x
)
=
lim
f
(
x
)
lim
g
(
x
)
=
A
B
.
\lim \frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}=\frac{A}{B}.
limg(x)f(x)=limg(x)limf(x)=BA.推论1 如果
lim
f
(
x
)
\lim f(x)
limf(x) 存在，而
c
c
c 为常数，那么
lim
[
c
f
(
x
)
]
=
c
lim
f
(
x
)
.
\lim [cf(x)]=c\lim f(x).
lim[cf(x)]=climf(x).推论2 如果
lim
f
(
x
)
\lim f(x)
limf(x) 存在，而
n
n
n 是正整数，那么
lim
[
f
(
x
)
]
n
=
[
lim
f
(
x
)
]
n
\lim [f(x)]^n=[\lim f(x)]^n
lim[f(x)]n=[limf(x)]n
定理4 设有数列
{
x
n
}
\{x_n\}
{xn} 和
{
y
n
}
\{y_n\}
{yn}.如果
lim
n
→
∞
x
n
=
A
,
lim
n
→
∞
y
n
=
B
,
\displaystyle\lim_{n\to \infty}x_n=A, \quad \displaystyle\lim_{n\to \infty}y_n=B,
n→∞limxn=A,n→∞limyn=B,那么
(
1
)
lim
n
→
∞
(
x
n
±
y
n
)
=
A
±
B
;
\quad (1) \displaystyle\lim_{n\to \infty}(x_n\pm y_n)=A\pm B;
(1)n→∞lim(xn±yn)=A±B;
(
2
)
lim
n
→
∞
(
x
n
⋅
y
n
)
=
A
⋅
B
;
\quad (2) \displaystyle\lim_{n\to \infty}(x_n\cdot y_n)=A\cdot B;
(2)n→∞lim(xn⋅yn)=A⋅B;
(
3
)
\quad (3)
(3) 当
y
n
≠
0
(
n
=
1
,
2
,
⋯
)
y_n\ne0 (n=1,2,\cdots)
yn=0(n=1,2,⋯) 且
B
≠
0
B\ne0
B=0 时，
lim
n
→
∞
x
n
y
n
=
A
B
\displaystyle\lim_{n\to \infty}\frac{x_n}{y_n}=\frac{A}{B}
n→∞limynxn=BA.
定理5 如果
φ
(
x
)
≥
ψ
(
x
)
\varphi(x)\ge\psi(x)
φ(x)≥ψ(x),而
lim
φ
(
x
)
=
A
,
lim
ψ
(
x
)
=
B
\lim \varphi(x)=A,\lim \psi(x)=B
limφ(x)=A,limψ(x)=B,那么
A
≥
B
.
A\ge B.
A≥B.
定理6(复合函数的极限运算法则)——设函数
y
=
f
[
g
(
x
)
]
y=f[g(x)]
y=f[g(x)] 是由函数
u
=
g
(
x
)
u=g(x)
u=g(x) 与函数
y
=
f
(
u
)
y=f(u)
y=f(u) 复合而成，
f
[
g
(
x
)
]
f[g(x)]
f[g(x)] 在点
x
0
x_0
x0 的某去心邻域内有定义，若
lim
x
→
x
0
g
(
x
)
=
u
0
,
lim
u
→
u
0
f
(
u
)
=
A
\displaystyle\lim_{x\to x_0}g(x)=u_0,\displaystyle\lim_{u\to u_0}f(u)=A
x→x0limg(x)=u0,u→u0limf(u)=A, 且存在
δ
0
>
0
,
\delta_0\gt 0,
δ0>0, 当
x
∈
U
˚
(
x
0
,
δ
0
)
x\in \mathring{U}(x_0,\delta_0)
x∈U˚(x0,δ0) 时，有
g
(
x
)
≠
u
0
,
g(x)\ne u_0,
g(x)=u0, 则
lim
x
→
x
0
f
[
g
(
x
)
]
=
lim
u
→
u
0
f
(
u
)
=
A
.
\displaystyle\lim_{x\to x_0}f[g(x)]=\displaystyle\lim_{u\to u_0}f(u)=A.
x→x0limf[g(x)]=u→u0limf(u)=A.
习题 1-5