次幂同时含有变量和常量的极限怎么计算？

一、题目

$$
I=\lim \limits_{x \rightarrow \infty} \frac{(x+a)^{x+a}(x+b)^{x+b}}{(x+a+b)^{2 x+a+b}} = ?
$$

难度评级：

二、解析

$$
\begin{aligned}
I= \\ \\
& \lim \limits_{x \rightarrow \infty} \frac{(x+a)^{x+a}}{(x+a+b)^{x+a}} \cdot \lim \limits_{x \rightarrow \infty} \frac{(x+b)^{x+b}}{(x+a+b)^{x+b}} = \\ \\
& \lim \limits_{x \rightarrow \infty} \frac{1}{\left(\frac{x+a+b}{x+a}\right)^{x+a}} \cdot \lim \limits_{x \rightarrow \infty} \frac{1}{\left(\frac{x+a+b}{x+b}\right)^{x+b}} = \\ \\
& \lim \limits_{x \rightarrow \infty} \frac{1}{\left(1+\frac{b}{x+a}\right)^{x+a}} \cdot \lim \limits_{x \rightarrow \infty} \frac{1}{\left(1+\frac{a}{x+b}\right)^{x+b}}= \\ \\
& \lim \limits_{x \rightarrow \infty} \frac{1}{\left(1+\frac{b}{x+a}\right)^{\frac{x+a}{b} \cdot \frac{b}{x+a} \cdot(x+a)}} \cdot \lim \limits_{x \rightarrow \infty} \frac{1}{\left(1+\frac{a}{x+b}\right)^{\frac{x+b}{a} \cdot \frac{a}{x+b} \cdot(x+b)}} = \\ \\
& \frac{1}{e^{b}} \cdot \frac{1}{e^{a}}
\end{aligned}
$$

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