# 函數極限

（重定向自函数极限

${\displaystyle x}$ ${\displaystyle {\frac {\sin x}{x}}}$
1 0.841471...
0.1 0.998334...
0.01 0.999983...

## 常用公式

### 有理函數

• ${\displaystyle \lim _{x\to \infty }{\frac {1}{x^{n}}}=0}$
• ${\displaystyle \lim _{x\to \infty }{\frac {1}{a^{x}}}=0}$
• ${\displaystyle \lim _{x\to 0^{+}}{1 \over x}=+\infty }$
• ${\displaystyle \lim _{x\to 0^{-}}{1 \over x}=-\infty .}$

### 無理函數

• ${\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{x}}=1}$
• ${\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}$

### 三角函數

• ${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}$
• ${\displaystyle \lim _{x\to \infty }{\frac {\sin x}{x}}=0}$

### 指數函數

• ${\displaystyle \lim _{x\to \infty }\left(1+{\frac {1}{x}}\right)^{x}=\lim _{x\to 0}(1+x)^{\frac {1}{x}}=e}$
• ${\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}$
• ${\displaystyle \lim _{x\to 0^{+}}x^{x}=1}$

### 對數函數

• ${\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}$
• ${\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty }$
• ${\displaystyle \lim _{x\to +\infty }\ln x=+\infty }$

## 參考

1. ^ 原文如下：On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist.