【数学分析笔记】第3章第3节无穷小量与无穷大量的阶(1)
3. 函数极限与连续函数
3.3 无穷小量与无穷大量的阶
3.3.1 无穷小量的阶
对于数列和函数来说,无穷小量的定义如下:
- 数列无穷小量: lim n → ∞ x n = 0 \lim\limits_{n\to\infty}x_{n}=0 n→∞limxn=0,则 { x n } \{x_{n}\} {xn}是无穷小量。
- 函数无穷小量:
lim
x
→
x
0
f
(
x
)
=
0
\lim\limits_{x\to x_{0}}f(x)=0
x→x0limf(x)=0,则称当
x
→
x
0
x\to x_{0}
x→x0时,
f
(
x
)
f(x)
f(x)是无穷小量。
【注】 x → x 0 x\to x_{0} x→x0可以换成 x → x 0 − , x → x 0 + , x → + ∞ , x → − ∞ , x → ∞ x\to x_{0}^{-},x\to x_{0}^{+},x\to +\infty,x\to -\infty,x\to \infty x→x0−,x→x0+,x→+∞,x→−∞,x→∞,一共六种情况。
当 x → x 0 x\to x_{0} x→x0, u ( x ) , v ( x ) u(x),v(x) u(x),v(x)都是无穷小量。
- 若
lim
x
→
x
0
u
(
x
)
v
(
x
)
=
0
\lim\limits_{x\to x_{0}}\frac{u(x)}{v(x)}=0
x→x0limv(x)u(x)=0,则称当
x
→
x
0
x\to x_{0}
x→x0,
u
(
x
)
u(x)
u(x)是
v
(
x
)
v(x)
v(x)的高阶无穷小量,
v
(
x
)
v(x)
v(x)是
u
(
x
)
u(x)
u(x)的低阶无穷小量,记为
u
(
x
)
=
o
(
v
(
x
)
)
,
(
x
→
x
0
)
u(x)=o(v(x)),(x\to x_{0})
u(x)=o(v(x)),(x→x0);
【例】 lim x → 0 1 − cos x x = lim x → 0 2 sin 2 x 2 x = lim x → 0 2 x sin 2 x 2 4 × ( x 2 ) 2 = 0 \lim\limits_{x\to 0}\frac{1-\cos x}{x}=\lim\limits_{x\to 0}\frac{2\sin ^{2}\frac{x}{2}}{x}=\lim\limits_{x\to 0}\frac{2x\sin ^{2}\frac{x}{2}}{4\times(\frac{x}{2})^{2}}=0 x→0limx1−cosx=x→0limx2sin22x=x→0lim4×(2x)22xsin22x=0,则当 x → 0 x\to 0 x→0时, 1 − cos x 1-\cos x 1−cosx是比 x x x的高阶无穷小量,即 1 − cos x = o ( x ) , ( x → 0 ) 1-\cos x=o(x),(x\to 0) 1−cosx=o(x),(x→0)
【例】 lim x → 0 tan x − sin x x 2 = lim x → 0 sin x cos x − sin x x 2 = lim x → 0 sin x cos x ⋅ 1 − cos x x 2 = lim x → 0 sin x x cos x ⋅ 1 − cos x x = lim x → 0 1 − cos x x = 0 \lim\limits_{x\to 0}\frac{\tan x-\sin x}{x^{2}}=\lim\limits_{x\to 0}\frac{\frac{\sin x}{\cos x}-\sin x}{x^{2}}=\lim\limits_{x\to 0}\frac{\sin x}{\cos x}\cdot\frac{1-\cos x}{x^{2}}=\lim\limits_{x\to 0}\frac{\sin x}{x\cos x}\cdot\frac{1-\cos x}{x}=\lim\limits_{x\to 0}\frac{1-\cos x}{x}=0 x→0limx2tanx−sinx=x→0limx2cosxsinx−sinx=x→0limcosxsinx⋅x21−cosx=x→0limxcosxsinx⋅x1−cosx=x→0limx1−cosx=0,则当 x → 0 x\to 0 x→0时, tan x − sin x = o ( x 2 ) , ( x → 0 ) \tan x-\sin x=o(x^{2}),(x\to 0) tanx−sinx=o(x2),(x→0) - 若存在
A
>
0
A>0
A>0,当
x
x
x在
x
0
x_{0}
x0的某一去心邻域中
{
x
∣
0
<
∣
x
−
x
0
∣
<
ρ
}
\{x|0<|x-x_{0}|<\rho\}
{x∣0<∣x−x0∣<ρ}成立
∣
u
(
x
)
v
(
x
)
∣
≤
A
|\frac{u(x)}{v(x)}|\le A
∣v(x)u(x)∣≤A,则称当
x
→
x
0
x\to x_{0}
x→x0时,
u
(
x
)
v
(
x
)
\frac{u(x)}{v(x)}
v(x)u(x)是有界量,记为
u
(
x
)
=
O
(
v
(
x
)
)
,
(
x
→
x
0
)
u(x)=O(v(x)),(x\to x_{0})
u(x)=O(v(x)),(x→x0)
【例】 x → 0 , u ( x ) = x sin 1 x , v ( x ) = x , ∣ u ( x ) v ( x ) ∣ = ∣ sin 1 x ∣ ≤ 1 x\to 0,u(x)=x\sin \frac{1}{x},v(x)=x,|\frac{u(x)}{v(x)}|=|\sin \frac{1}{x}|\le 1 x→0,u(x)=xsinx1,v(x)=x,∣v(x)u(x)∣=∣sinx1∣≤1,则 u ( x ) v ( x ) \frac{u(x)}{v(x)} v(x)u(x)是有界量,即 u ( x ) = O ( v ( x ) ) , ( x → 0 ) u(x)=O(v(x)),(x\to 0) u(x)=O(v(x)),(x→0) - 若
∃
0
<
a
<
A
<
+
∞
\exists 0<a<A<+\infty
∃0<a<A<+∞,在
x
0
x_{0}
x0的一个去心邻域
{
x
∣
0
<
∣
x
−
x
0
∣
<
ρ
}
\{x|0<|x-x_{0}|<\rho\}
{x∣0<∣x−x0∣<ρ}中,
a
≤
∣
u
(
x
)
v
(
x
)
∣
≤
A
<
+
∞
a\le |\frac{u(x)}{v(x)}|\le A<+\infty
a≤∣v(x)u(x)∣≤A<+∞,则称
u
(
x
)
,
v
(
x
)
u(x),v(x)
u(x),v(x)(当
x
→
x
0
x\to x_{0}
x→x0)是同阶无穷小量。
若 lim x → x 0 u ( x ) v ( x ) = c ≠ 0 \lim\limits_{x \rightarrow x_{0}} \frac{u(x)}{v(x)}=c \neq 0 x→x0limv(x)u(x)=c=0, u ( x ) u(x) u(x)与 v ( x ) v(x) v(x)必是同阶无穷小量。
【例】 u ( x ) = x ( 2 + sin 1 x ) , v ( x ) = x , ( x → 0 ) , ∣ u ( x ) v ( x ) ∣ = 0 < 1 ≤ ∣ 2 + sin 1 x ∣ ≤ 3 < + ∞ u(x)=x(2+\sin \frac{1}{x}),v(x)=x,(x\to 0),|\frac{u(x)}{v(x)}|=0<1\le|2+\sin\frac{1}{x}|\le 3<+\infty u(x)=x(2+sinx1),v(x)=x,(x→0),∣v(x)u(x)∣=0<1≤∣2+sinx1∣≤3<+∞, u ( x ) , v ( x ) u(x),v(x) u(x),v(x)是同阶无穷小量。 - 若
lim
x
→
x
0
u
(
x
)
v
(
x
)
=
1
\lim\limits_{x\to x_{0}}\frac{u(x)}{v(x)}=1
x→x0limv(x)u(x)=1,则称当
x
→
x
0
x\to x_{0}
x→x0时,
u
(
x
)
u(x)
u(x)与
v
(
x
)
v(x)
v(x)是等价无穷小量,记为
u
(
x
)
∼
v
(
x
)
,
(
x
→
x
0
)
u(x)\sim v(x),(x\to x_{0})
u(x)∼v(x),(x→x0),也记为
u
(
x
)
=
v
(
x
)
+
o
(
v
(
x
)
)
u(x)=v(x)+o(v(x))
u(x)=v(x)+o(v(x)).
【例】 lim x → 0 sin x x = 1 \lim\limits_{x\to 0}\frac{\sin x}{x}=1 x→0limxsinx=1,即 sin x ∼ x , ( x → 0 ) \sin x\sim x,(x\to 0) sinx∼x,(x→0),亦即 sin x = x + o ( x ) , ( x → 0 ) \sin x=x+o(x),(x\to 0) sinx=x+o(x),(x→0),同理 tan x ∼ x , ( x → 0 ) \tan x\sim x,(x\to 0) tanx∼x,(x→0)
【例】 lim x → 0 1 − cos x 1 2 x 2 = lim x → 0 2 sin 2 x 2 4 × 1 2 ⋅ ( x 2 ) 2 = 1 \lim\limits_{x\to 0}\frac{1-\cos x}{\frac{1}{2}x^{2}}=\lim\limits_{x\to 0}\frac{2\sin^{2}\frac{x}{2}}{4\times\frac{1}{2}\cdot(\frac{x}{2})^{2}}=1 x→0lim21x21−cosx=x→0lim4×21⋅(2x)22sin22x=1,即 1 − cos x ∼ 1 2 x 2 , ( x → 0 ) 1-\cos x\sim\frac{1}{2}x^{2},(x\to 0) 1−cosx∼21x2,(x→0),亦即 1 − cos x = 1 2 x 2 + o ( x 2 ) , ( x → 0 ) 1-\cos x=\frac{1}{2}x^{2}+o(x^{2}),(x\to 0) 1−cosx=21x2+o(x2),(x→0)(当 x → 0 x\to 0 x→0时, 1 − cos x − 1 2 x 2 1-\cos x-\frac{1}{2}x^{2} 1−cosx−21x2是 x 2 x^{2} x2的高阶无穷小)
【例】 lim x → 0 tan x − sin x 1 2 x 3 = lim x → 0 sin x cos x − sin x 1 2 x 3 = lim x → 0 sin x cos x ⋅ 1 − cos x 1 2 x 3 = lim x → 0 sin x x cos x ⋅ 1 − cos x 1 2 x 2 = 1 \lim\limits_{x\to 0}\frac{\tan x-\sin x}{\frac{1}{2}x^{3}}=\lim\limits_{x\to 0}\frac{\frac{\sin x}{\cos x}-\sin x}{\frac{1}{2}x^{3}}=\lim\limits_{x\to 0}\frac{\sin x}{\cos x}\cdot\frac{1-\cos x}{\frac{1}{2}x^{3}}=\lim\limits_{x\to 0}\frac{\sin x}{x\cos x}\cdot\frac{1-\cos x}{\frac{1}{2}x^{2}}=1 x→0lim21x3tanx−sinx=x→0lim21x3cosxsinx−sinx=x→0limcosxsinx⋅21x31−cosx=x→0limxcosxsinx⋅21x21−cosx=1,即 tan x − sin x ∼ 1 2 x 3 , ( x → 0 ) \tan x-\sin x\sim \frac{1}{2}x^{3},(x\to 0) tanx−sinx∼21x3,(x→0),亦即 tan x − sin x = 1 2 x 3 + o ( x 3 ) \tan x- \sin x=\frac{1}{2}x^{3}+o(x^{3}) tanx−sinx=21x3+o(x3)(和上面类似) - 若 lim x → x 0 u ( x ) v ( x ) = ∞ \lim\limits_{x\to x_{0}}\frac{u(x)}{v(x)}=\infty x→x0limv(x)u(x)=∞,则称当 x → x 0 x\to x_{0} x→x0, u ( x ) u(x) u(x)是 v ( x ) v(x) v(x)的低阶无穷小量, v ( x ) v(x) v(x)是 u ( x ) u(x) u(x)的高阶无穷小量,记为 v ( x ) = o ( u ( x ) ) , ( x → x 0 ) v(x)=o(u(x)),(x\to x_{0}) v(x)=o(u(x)),(x→x0);
- 特别地,若 lim x → x 0 + f ( x ) ( x − x 0 ) k = c ≠ 0 \lim\limits_{x \rightarrow x_{0}^{+}} \frac{f(x)}{\left(x-x_{0}\right)^{k}}=c \neq 0 x→x0+lim(x−x0)kf(x)=c=0时( k > 0 k>0 k>0为常数),则称 f ( x ) f(x) f(x)当 x → x 0 + x\to x_{0}^{+} x→x0+时是 ( x − x 0 ) (x-x_{0}) (x−x0)的 k k k阶无穷小量。
- 若
lim
x
→
x
0
f
(
x
)
x
k
=
c
≠
0
\lim\limits_{x \rightarrow x_{0}} \frac{f(x)}{x^{k}}=c \neq 0
x→x0limxkf(x)=c=0时(
k
>
0
k>0
k>0为常数),则称
f
(
x
)
f(x)
f(x)当
x
→
x
0
x\to x_{0}
x→x0时是
x
x
x的
k
k
k阶无穷小量。
【注】1)取 v ( x ) = ( x − x 0 ) k v(x)=(x-x_{0})^{k} v(x)=(x−x0)k可知 u ( x ) u(x) u(x)是几阶无穷小量;
2) lim x → 0 + − 1 ln x \lim\limits_{x\to 0^{+}}\frac{-1}{\ln x} x→0+limlnx−1是无穷小量。发现对任意的 α > 0 , − 1 ln x \alpha>0,\frac{-1}{\ln x} α>0,lnx−1是 x α x^{\alpha} xα低阶无穷小量, lim x → 0 + − 1 ln x x α = − ∞ \lim\limits_{x\to 0^{+}}\frac{\frac{-1}{\ln x}}{x^{\alpha}}=-\infty x→0+limxαlnx−1=−∞,记 − 1 ln x = o ( 1 ) , ( x → 0 ) \frac{-1}{\ln x}=o(1),(x\to 0) lnx−1=o(1),(x→0)(比1阶无穷小量的阶还小,算不出来是几阶无穷小也可以写 o ( 1 ) o(1) o(1))
【例】 x → 0 , u ( x ) = sin 1 x x\to 0,u(x)=\sin\frac{1}{x} x→0,u(x)=sinx1,记为 u ( x ) = O ( 1 ) , ( x → 0 ) u(x)=O(1),(x\to 0) u(x)=O(1),(x→0)(即这个量本身有界)
3.3.2 无穷大量的阶
lim
x
→
x
0
f
(
x
)
=
∞
(
±
∞
)
\lim\limits_{x\to x_{0}}f(x)=\infty(\pm\infty)
x→x0limf(x)=∞(±∞),则称当
x
→
x
0
x\to x_{0}
x→x0时,
f
(
x
)
f(x)
f(x)是(正、负)无穷大量。
【注】
x
→
x
0
x\to x_{0}
x→x0可以换成
x
→
x
0
−
,
x
→
x
0
+
,
x
→
+
∞
,
x
→
−
∞
,
x
→
∞
x\to x_{0}^{-},x\to x_{0}^{+},x\to +\infty,x\to -\infty,x\to \infty
x→x0−,x→x0+,x→+∞,x→−∞,x→∞,一共六种情况。
设
u
(
x
)
,
v
(
x
)
u(x),v(x)
u(x),v(x),当
x
→
x
0
x\to x_{0}
x→x0时都是无穷大量。
- lim x → x 0 u ( x ) v ( x ) = ∞ \lim\limits_{x\to x_{0}}\frac{u(x)}{v(x)}=\infty x→x0limv(x)u(x)=∞,说明当 x → x 0 x\to x_{0} x→x0, u ( x ) u(x) u(x)是 v ( x ) v(x) v(x)的高阶无穷大量, v ( x ) v(x) v(x)是 u ( x ) u(x) u(x)的低阶无穷大量。
-
lim
x
→
x
0
u
(
x
)
v
(
x
)
=
0
\lim\limits_{x\to x_{0}}\frac{u(x)}{v(x)}=0
x→x0limv(x)u(x)=0,说明当
x
→
x
0
x\to x_{0}
x→x0,
u
(
x
)
u(x)
u(x)是
v
(
x
)
v(x)
v(x)的低阶无穷大量,
v
(
x
)
v(x)
v(x)是
u
(
x
)
u(x)
u(x)的高阶无穷大量。
无穷大量有如下大小关系
n n > > n ! > > a n ( a > 1 ) > > n α ( α > 0 ) > > ln β n ( β > 0 ) n^{n}>>n!>>a^{n}(a>1)>>n^{\alpha}(\alpha>0)>>\ln^{\beta}n(\beta >0) nn>>n!>>an(a>1)>>nα(α>0)>>lnβn(β>0)
(剩下内容下节课再记)
原文地址:https://blog.csdn.net/qq_30204431/article/details/142374075
免责声明:本站文章内容转载自网络资源,如本站内容侵犯了原著者的合法权益,可联系本站删除。更多内容请关注自学内容网(zxcms.com)!