# 極限の一覧

## 一般的な極限の性質

${\displaystyle {\mbox{If }}\lim _{x\to c}f(x)=L_{1}{\mbox{ and }}\lim _{x\to c}g(x)=L_{2}{\mbox{ then:}}}$
${\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}$
${\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}}$
${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\mbox{ if }}L_{2}\neq 0}$
${\displaystyle \lim _{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad {\mbox{ if }}n{\mbox{ is a positive integer}}}$
${\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L_{1}^{1 \over n}\qquad {\mbox{ if }}n{\mbox{ is a positive integer, and if }}n{\mbox{ is even, then }}L_{1}>0}$
${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}\qquad {\mbox{ if }}\lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\mbox{ or }}\lim _{x\to c}|g(x)|=+\infty }$ (ロピタルの定理)

## 単純な関数

${\displaystyle \lim _{x\to c}a=a}$
${\displaystyle \lim _{x\to c}x=c}$
${\displaystyle \lim _{x\to c}(ax+b)=ac+b}$
${\displaystyle \lim _{x\to c}x^{r}=c^{r}\qquad {\mbox{ if }}r{\mbox{ is a positive integer}}}$
${\displaystyle \lim _{x\to +0}{\frac {1}{x^{r}}}=+\infty }$
${\displaystyle \lim _{x\to -0}{\frac {1}{x^{r}}}=\left\{{\begin{matrix}-\infty ,&{\mbox{if }}r{\mbox{ is odd}}\\+\infty ,&{\mbox{if }}r{\mbox{ is even}}\end{matrix}}\right.}$

## 対数関数と指数関数

${\displaystyle \lim _{x\to +0}\log _{a}x={\begin{cases}-\infty ,&a>1\\\infty ,&a<1\end{cases}}}$
${\displaystyle \lim _{x\to -\infty }a^{x}=0\qquad {\mbox{ if }}a>1}$

## 三角関数

${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}$
${\displaystyle \lim _{x\to a}\sin x=\sin a}$
${\displaystyle \lim _{x\to a}\cos x=\cos a}$
${\displaystyle \lim _{x\to n\pm 0}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad {\mbox{ for any integer }}n}$

## その他の諸関数

${\displaystyle \lim _{x\to \infty }{\frac {N}{x}}=0{\mbox{ for any real number }}N}$
${\displaystyle \lim _{x\to \infty }{\frac {x}{N}}={\begin{cases}\infty ,&N>0\\{\mbox{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&N<1\end{cases}}}$
${\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }1/N^{x}=0{\mbox{ for any }}N>1}$
${\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\mbox{does not exist}},&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty {\mbox{ for any positive integer }}N}$
${\displaystyle \lim _{x\to \infty }\log x=\infty }$
${\displaystyle \lim _{x\to +0}\log x=-\infty }$

## 備考

• positive - 正の
• integer - 整数
• even - 偶数の
• odd - 奇数の
• any - 任意の
• real - 実数の
• does not exist - 存在せず