# 不定式 (數學)

（重定向自不定式 (数学)

## 例子

0除以0

${\displaystyle {\frac {0}{0}}}$  是不定式。

0的0次方

${\displaystyle 0^{0}}$ 也是不定式。在不同軟件中，有不同的處理規則，有些定義為1，有些視為「沒有定義」。

${\displaystyle \lim _{x\to 0^{+}}0^{x}=0\qquad }$
${\displaystyle \lim _{x\to 0^{+}}x^{0}=1\qquad }$
${\displaystyle \lim _{x\to 0^{+}}x^{x}=1\qquad }$

${\displaystyle (1+x)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}}$

${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}$

## 物理

${\displaystyle \lim _{x\to c}{f(x) \over g(x)}}$  ${\displaystyle f(c)=g(c)=0\,}$ 。若 ${\displaystyle f(x)\,}$  等于 ${\displaystyle g(x)\,}$ ，极限为一；若 ${\displaystyle f(x)\,}$  等于 ${\displaystyle g(x)\,}$ 两倍，则极限为二。

## 不定式列表

${\displaystyle 0/0}$  ${\displaystyle \lim _{x\to c}f(x)=0,\ \lim _{x\to c}g(x)=0\!}$
${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {1/g(x)}{1/f(x)}}\!}$
${\displaystyle \infty /\infty }$  ${\displaystyle \lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=\infty \!}$  ${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {1/g(x)}{1/f(x)}}\!}$
${\displaystyle 0\cdot \infty }$  ${\displaystyle \lim _{x\to c}f(x)=0,\ \lim _{x\to c}g(x)=\infty \!}$  ${\displaystyle \lim _{x\to c}f(x)g(x)=\lim _{x\to c}{\frac {f(x)}{1/g(x)}}\!}$  ${\displaystyle \lim _{x\to c}f(x)g(x)=\lim _{x\to c}{\frac {g(x)}{1/f(x)}}\!}$
${\displaystyle \infty -\infty }$  ${\displaystyle \lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=\infty \!}$  ${\displaystyle \lim _{x\to c}(f(x)-g(x))=\lim _{x\to c}{\frac {1/g(x)-1/f(x)}{1/(f(x)g(x))}}\!}$  ${\displaystyle \lim _{x\to c}(f(x)-g(x))=\ln \lim _{x\to c}{\frac {e^{f(x)}}{e^{g(x)}}}\!}$
${\displaystyle 0^{0}}$  ${\displaystyle \lim _{x\to c}f(x)=0^{+},\lim _{x\to c}g(x)=0\!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!}$
${\displaystyle 1^{\infty }}$  ${\displaystyle \lim _{x\to c}f(x)=1,\ \lim _{x\to c}g(x)=\infty \!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!}$
${\displaystyle \infty ^{0}}$  ${\displaystyle \lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=0\!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!}$  ${\displaystyle \lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!}$