左极限与右极限

castelu

定义　设函数$f$在${U^\circ}_+ \left( x_0;\delta ' \right)$（或${U^\circ}_- \left( x_0;\delta ' \right)$内有定义，$A$为定数。若对任给的$\epsilon > 0$，存在正数$\delta(<\delta')$，使得当$x_0 < x < x_0 + \delta$（或$x_0 - \delta < x < x_0$）时有
$$\left| f(x) - A \right| < \epsilon，$$
　　则称数$A$为函数$f$当$x$趋于$x_0^+$（或$x_0^-$）时的右（左）极限，记作
$$\lim\limits_{x \rightarrow x_0^+}f\left( x \right) = A\left( \lim\limits_{x \rightarrow x_0^-}f\left( x \right) = A \right)$$
　　或
$$f\left( x \right) \to A (x \rightarrow x_0^+)\left( f\left( x \right) \rightarrow A (x \to x_0^-) \right)。$$
　　右极限与左极限统称为单侧极限。$f$在点$x_0$的右极限与左极限又分别记为
$$f\left( x_0+0 \right) = \lim\limits_{x \rightarrow x_0^+}f\left( x \right)与f\left( x_0-0 \right) = \lim\limits_{x \rightarrow x_0^-}f\left( x \right)。$$

定理　$\lim\limits_{x \to x_0}f\left( x \right) = A \Leftrightarrow \lim\limits_{x \rightarrow x_0^+}f\left( x \right)=\lim\limits_{x \to x_0^-}f\left( x \right)=A$。