極限の一覧

極限の一覧は、解析学における代表的な関数の極限の一覧である。極限に関しては極限の項を参照のこと。 以下で、xは変数、a、b、cは定数である。 If  lim x → c f ( x ) = L 1  and  lim x → c g ( x ) = L 2  then: {\displaystyle {\mbox{If }}\lim _{x\to c}f(x)=L_{1}{\mbox{ and }}\lim _{x\to c}g(x)=L_{2}{\mbox{ then:}}} lim x → c [ f ( x ) ± g ( x ) ] = L 1 ± L 2 {\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}} lim x → c [ f ( x ) g ( x ) ] = L 1 × L 2 {\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}} lim x → c f ( x ) g ( x ) = L 1 L 2  if  L 2 ≠ 0 {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\mbox{ if }}L_{2}\neq 0} lim x → c f ( x ) n = L 1 n  if  n  is a positive integer {\displaystyle \lim _{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad {\mbox{ if }}n{\mbox{ is a positive integer}}} lim x → c f ( x ) 1 n = L 1 1 n  if  n  is a positive integer, and if  n  is even, then  L 1 > 0 {\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} lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x )  if  lim x → c f ( x ) = lim x → c g ( x ) = 0  or  lim x → c | g ( x ) | = + ∞ {\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) lim x → c a = a {\displaystyle \lim _{x\to c}a=a} lim x → c x = c {\displaystyle \lim _{x\to c}x=c} lim x → c ( a x + b ) = a c + b {\displaystyle \lim _{x\to c}(ax+b)=ac+b} lim x → c x r = c r  if  r  is a positive integer {\displaystyle \lim _{x\to c}x^{r}=c^{r}\qquad {\mbox{ if }}r{\mbox{ is a positive integer}}} lim x → + 0 1 x r = + ∞ {\displaystyle \lim _{x\to +0}{\frac {1}{x^{r}}}=+\infty } lim x → − 0 1 x r = { − ∞ , if  r  is odd + ∞ , if  r  is even {\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.} lim x → + 0 log a ⁡ x = { − ∞ , a > 1 ∞ , a < 1 {\displaystyle \lim _{x\to +0}\log _{a}x={\begin{cases}-\infty ,&a>1\\\infty ,&a<1\end{cases}}} lim x → − ∞ a x = 0  if  a > 1 {\displaystyle \lim _{x\to -\infty }a^{x}=0\qquad {\mbox{ if }}a>1} lim x → 0 sin ⁡ x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1} lim x → a sin ⁡ x = sin ⁡ a {\displaystyle \lim _{x\to a}\sin x=\sin a} lim x → a cos ⁡ x = cos ⁡ a {\displaystyle \lim _{x\to a}\cos x=\cos a} lim x → n ± 0 tan ⁡ ( π x + π 2 ) = ∓ ∞  for any integer  n {\displaystyle \lim _{x\to n\pm 0}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad {\mbox{ for any integer }}n} lim x → ∞ N x = 0  for any real number  N {\displaystyle \lim _{x\to \infty }{\frac {N}{x}}=0{\mbox{ for any real number }}N} lim x → ∞ x N = { ∞ , N > 0 does not exist , N = 0 − ∞ , N < 0 {\displaystyle \lim _{x\to \infty }{\frac {x}{N}}={\begin{cases}\infty ,&N>0\\{\mbox{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}} lim x → ∞ x N = { ∞ , N > 0 1 , N = 0 0 , N < 0 {\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}} lim x → ∞ N x = { ∞ , N > 1 1 , N = 1 0 , N < 1 {\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&N<1\end{cases}}} lim x → ∞ N − x = lim x → ∞ 1 / N x = 0  for any  N > 1 {\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }1/N^{x}=0{\mbox{ for any }}N>1} lim x → ∞ N x = { 1 , N > 0 0 , N = 0 does not exist , N < 0 {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\mbox{does not exist}},&N<0\end{cases}}} lim x → ∞ x N = ∞  for any positive integer  N {\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty {\mbox{ for any positive integer }}N} lim x → ∞ log ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log x=\infty } lim x → + 0 log ⁡ x = − ∞ {\displaystyle \lim _{x\to +0}\log x=-\infty } 上記に使われた用語の和訳を以下に示す。 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 極限