等价无穷小代换定理

castelu

定理　设函数$f$，$g$，$h$在$U^\circ \left( x_0 \right)$内有定义，且有
$$f\left( x \right) \sim g\left( x \right)\left( x \to x_0 \right)$$
（i）若$\lim\limits_{x \rightarrow x_0}f\left( x \right)h\left( x \right) = A$，则$\lim\limits_{x \rightarrow x_0}g\left( x \right)h\left( x \right) = A$；
（ii）若$\lim\limits_{x \rightarrow x_0}\frac{h\left( x \right)}{f\left( x \right)} = B$，则$\lim\limits_{x \rightarrow x_0}\frac{h\left( x \right)}{g\left( x \right)} = B$。