了解了实数理论,可以对极限理论进行复习.
一.极限的定义.
1.序列的极限:$\{x_n\}$,$\lim_{n \rightarrow \infty}{x_n}=a \Leftrightarrow \forall \epsilon>0$,$\exists N$,当$n>N$时,$|x_n-a|<\epsilon$.
当$a=0$时,称$\{x_n\}$为无穷小量.故无穷小量不是一个数,而是以0为极限的序列.
以下说法错误:
(1)$n$越大时,$x_n-a$越小,则$\lim_{n \rightarrow \infty}{x_n}=a$.反例$\{-n\}$.
(2)当$n$越大时,$|x_n-a|$越来越向0靠拢,则$\lim_{n \rightarrow \infty}{x_n}=a$.$x_n = 2 + \frac{1}{n}$,$a=1$.
(3)$\forall \epsilon>0$,$\exists N$,当$n>N$时,有无穷多个$x_n$满足$|x_n-a|<\epsilon$,则$\lim_{n \rightarrow \infty}{x_n}=a$.取
\[x_n = \begin{cases}1, n\text{为偶数}\\\frac{1}{n},n\text{为奇数}\end{cases}\]
(4)$\forall N>0$,$\exists \epsilon$,使$n>N$时,$|x_n-a|<\epsilon$.
\[x_n = \begin{cases}1, n\text{为偶数}\\-1,n\text{为奇数}\end{cases}\]
2.函数的极限.(1)$\lim\limits_{x \rightarrow +\infty}{f(x)}=A \Leftrightarrow \forall \epsilon$,$\exists M>0$,使$x>M$时有$|f(x)-A|<\epsilon$.这里$f(x)$定义在$[a,+\infty)$上.
(2)$\lim\limits_{x \rightarrow x_0}{f(x)}=A \Leftrightarrow \forall \epsilon$,$\exists \delta>0$,当$0<|x-x_0|<\delta$时,有$|f(x)-A|<\epsilon$.这里$f(x)$定义在$O(x_0,\delta’)$上.
同(1)类似,可定义$\lim\limits_{x \rightarrow -\infty}{f(x)}$,$\lim\limits_{x \rightarrow \infty}{f(x)}$.
二.极限的性质.
1(唯一性).若极限$\lim\limits_{x \rightarrow x_0}{f(x)}$或$\lim\limits_{n \rightarrow \infty}{f(x)}$存在,则它只有一个极限.(反证)
2(有界性).$a_n$收敛$\Rightarrow$$\{a_n\}$有界.$\Leftarrow$不成立.
对于$f(x)$,$\lim\limits_{x \rightarrow x_0}{f(x)}$存在$\Rightarrow$$\exists U(x_0)$,使得$f(x)$在$U(x_0)$有界.
3(保号性).$\{a_n\}$收敛,且$\lim\limits_{n \rightarrow \infty}{a_n}=a>0$(或$<0$),则对任意$a>a’>0$(或$0>a’>a$)的$a’$,都存在$N>0$,使$n>N$时,$a_n>a’>0$(或$a_n<a'<0$).[根据定义,注意$\epsilon$的取值即可]
对于$f(x)$:$\lim\limits_{x \rightarrow x_0}{f(x)}=A>0$(或$<0$),则$\forall r>0$($r<|A|$),$\exists U(x_0)$,使得对于$x \in U(x_0)$,恒有$f(x)>r>0$(或$f(x)<-r<0$).
4(迫敛性).$\{a_n\}$,$\{b_n\}$收敛,且$\lim\limits_{n \rightarrow \infty}{a_n}=\lim\limits_{n \rightarrow \infty}{b_n}=a$,若存在$N_0$,当$n>N_0$时有$a_n \le c_n \le b_n$,则$\lim\limits_{n \rightarrow \infty}{c_n}=a$.
对于函数:若$\lim\limits_{x \rightarrow x_0}{f(x)}=\lim_{x \rightarrow x_0}{g(x)}=A$,$\exists U(x_0,\delta’)$,使得$\forall x \in U(x_0,\delta’)$,$f(x) \le h(x) \le g(x)$,则$\lim\limits_{x \rightarrow x_0}{h(x)}=A$.[根据定义即可]
5(不等式).$\{a_n\}$,$\{b_n\}$收敛,且对于$N_0 \in N$,当$n>N_0$时,有$a_n \le b_n$,则$\lim\limits_{n \rightarrow \infty}{a_n} \le \lim\limits_{n \rightarrow \infty}{b_n}$.
对于函数:若$\lim\limits_{x \rightarrow x_0}{f(x)}$,$\lim\limits_{x \rightarrow x_0}{g(x)}$存在,且$\exists U(x_0,\delta’)$,对$\forall x \in U(x_0,\delta’)$,有$f(x) \le g(x)$,则$\lim\limits_{x \rightarrow x_0}{f(x)} \le \lim\limits_{x \rightarrow x_0}{g(x)}$.[根据定义]
三.极限的运算法则:
若$\{a_n\}$,$\{b_n\}$收敛.则$\{a_n+b_n\}$,$\{a_n-b_n\}$,$a_n \cdot b_n$皆收敛.且
\begin{gather*}\lim{(a_n \pm b_n)} = \lim{a_n} \pm \lim{b_n}\\\lim{(a_nb_n)} = \lim{a_n} \cdot \lim{b_n}\end{gather*}
若$b_n \neq 0$,及$\lim{b_n} \neq 0$,则$\{\frac{a_n}{b_n}\}$收敛,且
\[\lim{\frac{a_n}{b_n}} = \frac{\lim{a_n}}{\lim{b_n}}.\]
对于函数有类似的性质.
四.极限存在的判定.
先引入几个概念.
(1)对于数列$\{a_n\}$,$\{n_k\}$是自然数集$N$的无限子集,且$n_1<n_2<\cdots<n_k<\cdots$,则$\{a_{n_k}\}$称为$\{a_n\}$的子列.$k=1,2,\cdots$.
(2)单侧极限.设函数$f(x)$在$U_+(x_0,\delta’)$[或$U_-(x_0,\delta’)$]内有定义,$A$是确定的数,$\forall \epsilon$,$\exists \delta>0$($\delta<\delta’$),使$x_0<x<x_0+\delta$($x_0-\delta<x<x_0$)时,有$|f(x)-A|<\epsilon$,则称函数$f$在$x$趋于$x_0+$($x_0-$)时右(左)极限存在.记作:
\[\lim_{x \rightarrow x_0+}{f(x)}=A \quad (\lim_{x \rightarrow x_0-}{f(x)}=A)\]
(3)上极限,下极限.
若在点(数)$a$的任一邻域内含有点列(数列)$\{x_n\}$的无限多个项,则称点(数)$a$为点列(数列)$\{x_n\}$的聚点.(注意,$x_n$中可以出现相同的数)
有界点列(数列)$\{x_n\}$至少有一个聚点,且存在最大聚点与最小聚点.(证明见课本)
有界点列(数列)$\{x_n\}$的最大聚点$\bar{A}$,称为点列(数列)$\{x_n\}$的上极限,最小聚点$\underline{A}$,称为$\{x_n\}$的下极限,记作:
\[\bar{A}=\varlimsup_{n \rightarrow \infty}{x_n} \quad \underline{A}=\varliminf_{n \rightarrow \infty}{x_n}.\]
几点性质:
(i)$\varliminf{x_n} \le \varlimsup{x_n}$.
(ii)$\varlimsup{x_n}=\bar{A}$ $\Leftrightarrow$ $\forall \alpha \in R$,当$\alpha>\bar{A}$时,$\{x_n\}$中大于$\alpha$的项至多有限个,$\alpha<\bar{A}$时,$\{x_n\}$中大于$\alpha$的项有无限多个.
(iii)$\varliminf{x_n}=\underline{A}$ $\Leftrightarrow$ $\forall \beta \in R$,$\beta < \underline{A}$时,$\{x_n\}$中小于$\beta$的项至多有限个,$\beta>\underline{A}$时,$\{x_n\}$中小于$\beta$的项有无限多个.
(iv)$\{x_n\}$为有界数列,则$\bar{A}$为$\{x_n\}$上极限 $\Leftrightarrow$ $\bar{A}=\lim\limits_{n \rightarrow \infty}{\sup\limits_{k \le n}{\{x_k\}}}$;$\underline{A}$为下极限$\Leftrightarrow$$\underline{A} = \lim\limits_{n \rightarrow \infty}{\inf\limits_{k \ge n}{\{x_k\}}}$.
判定极限存在的定理:
1.$\{a_n\}$收敛 $\Leftrightarrow$ $\{a_n\}$的任一子列收敛,且有相同的极限.
2.单调有界定理.
3.柯西收敛准则.
4.$\lim\limits_{x \rightarrow x_0}{f(x)}=A$ $\Leftrightarrow$ $\lim\limits_{x \rightarrow x_0+}{f(x)}=\lim\limits_{x \rightarrow x_0-}{f(x)}=A$.
5.Heine归结原则:$f$定义在$x_0$的某空心邻域$U(x_0)$,$\lim\limits_{x \rightarrow x_0}{f(x)}$存在$\Leftrightarrow$对任何以$x_0$为极限且含于$U(x_0)$的数列$\{x_n\}$极限$\lim\limits_{n \rightarrow \infty}{f(x_n)}$存在且相等.(必要性用定义,充分性用反证法)
6.函数极限的柯西准则:把$m,n>M$改为$x’,x” \in U(x_0,\delta)$即可.
极限不存在的充要条件:$\exists \epsilon_0$,$\forall \delta$,$\exists x’,x” \in U(x_0,\delta)$,使得$|f(x’)-f(x”)|>\epsilon_0$.
7.$\lim\limits_{n \rightarrow \infty}{x_n}=A$ $\Leftrightarrow$ $\varlimsup\limits_{n \rightarrow \infty}{x_n}=\varliminf\limits_{n \rightarrow \infty}{x_n}=A$.
五.极限的计算.
1.利用定义求极限;
2.用Cauchy准则来判断有否极限;
3.利用极限运算及已知的极限来求极限;
4.利用不等式求极限;
5.利用变量替换法求极限;
6.利用两个重要极限来求极限;
7.利用单调有界定理来求极限;
8.利用函数连续的性质求极限;
9.用洛必达(L’Hospital)法则求极限;
10.利用泰勒(Taylor)公式求极限;
11.通过阶的比较来求极限(用到Taylor公式)
两个重要极限:
\begin{gather*}\lim_{x \rightarrow 0}{\frac{\sin{x}}{x}} = 1\\\lim_{n \rightarrow \infty}{(1 + \frac{1}{n})^n}=e, \quad \lim_{x \rightarrow \infty}{(1 + \frac{1}{x})^x}=e.\end{gather*}
例子:
1.$\lim\limits_{x \rightarrow 1}{\frac{\sqrt[m]{x}-1}{\sqrt[n]{x}-1}}$($m,n \in N$)
令$y=x^{\frac{1}{mn}}$.
2.$\lim\limits_{x \rightarrow 0}{\frac{\sqrt[m]{1+x}-1-\frac{1}{m}x}{x^2}}$.
令$\sqrt[m]{1+x}-1=y$.
3.(1)$f(x)$在$x=a$连续,若$\lim\limits_{n \rightarrow \infty}{x_n}=a$,则$\lim\limits_{n \rightarrow \infty}{f(x_n)}=f(\lim\limits_{n \rightarrow \infty}{x_n})=f(a)$及$\lim\limits_{x \rightarrow a}{f(x)}=f(a)$.
(2)设$f(y)$在$y=b$连续,$y=g(x)$有$\lim\limits_{x \rightarrow a}{g(x)}=b$,则
\[\lim_{x \rightarrow a}{f[g(x)]} = f[\lim_{x \rightarrow a}{g(x)}]=f(b).\]
(3)设$\lim\limits_{x \rightarrow a}{u(x)}=A>0$,$\lim\limits_{x \rightarrow a}{v(x)}=B$,则
\[\lim_{x \rightarrow a}{u(x)^{v(x)}} = \lim_{x \rightarrow a}{e^{v(x)\ln{(u(x))}}}=e^{B\ln{A}}=A^B.\]
六.阶的估计(更详细的见潘承洞,于秀源的《阶的估计》)
1.若$\lim\limits_{x \rightarrow x_0}{\frac{f(x)}{g(x)}}=0$,则称$f(x)$对$g(x)$当$x \rightarrow x_0$时是无穷小量,记作$f(x)=o(g(x)), x \rightarrow x_0$.
$f(x)$,$g(x)$都是无穷大量,则称$f(x)$是比$g(x)$低阶的无穷大量;
$f(x)$,$g(x)$都是无穷小量,则称$f(x)$是比$g(x)$高阶的无穷小量;
若$\lim\limits_{x \rightarrow x_0}{\frac{f(x)}{g(x)}}=1$.则称$f(x)$与$g(x)$当$x \rightarrow x_0$时是等价的.记作:$f(x) \sim g(x)$,$x \rightarrow x_0$.
设$g(x)>0$,若存在常数$A>0$,使$|f(x)| \le Ag(x)$,$x\in [a,b]$成立,则称$g(x)$是$f(x)$的强函数,记为$f(x)=O(g(x))$,$x \in [a,b]$.
设当$x \rightarrow x_0$时,$f(x)$与$g(x)$都是无穷大量(小量),且存在常数$A>0$,$B>0$,使得$\varlimsup\limits_{x \rightarrow x_0}{\frac{f(x)}{g(x)}} \le A$,$\varlimsup\limits_{x \rightarrow x_0}{\frac{g(x)}{f(x)}} \le B$,则称$f(x)$与$g(x)$当$x \rightarrow x_0$时是同阶无穷大量(小量).常数$A$,$B$称为大$O$常数.
2.几点性质:
(1)若$f(x)$是无穷大量,$x \rightarrow x_0$,而$\varphi(x)=O(1)$,则$\varphi(x)=o(f(x))$,$x \rightarrow x_0$;
(2)若$f(x)=O(\varphi)$,$\varphi=O(\psi)$,则$f(x)=O(\psi)$;
(3)若$f(x)=O(\varphi)$,$\varphi=o(\psi)$,则$f(x)=o(\psi)$;
(4)$O(f)+O(g)=O(f+g)$;
(5)$O(f)+O(g)=O(fg)$;
(6)$o(1)O(f)=o(f)$;
(7)$O(1)o(f)=o(f)$;
(8)$O(f)+o(f)=O(f)$;
(9)$o(f)+o(g)=o(|f|+|g|)$;
(10)$o(f) \cdot o(g)=o(fg)$;
(11)$\{O(f)\}^k=O_k(f^k)$,$k \in N$;
(12)$\{o(f)\}^k=o(f^k)$;
(13)$f \sim g$,$g \sim \varphi$ $\Rightarrow$ $f \sim \varphi$;
(14)$f=o(g)$,$g \sim \varphi$ $\Rightarrow$ $g = \varphi \pm f$;
$O_k$表示大$O$常数与$k$有关.这些证明根据定义即可,较易.