📝 题目
例 5 设 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = a}$ ,求证: $\mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( 1 + \frac{{x}_{n}}{n}\right) }^{n} = {\mathrm{e}}^{a}$ .
💡 答案与解析
证 令 ${y}_{n}\frac{\text{ 定义 }n}{{x}_{n}}$ ,显然有 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{y}_{n} = \infty}$ . 于是
$$ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( 1 + \frac{{x}_{n}}{n}\right) }^{n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left\lbrack {\left( 1 + \frac{1}{{y}_{n}}\right) }^{{y}_{n}}\right\rbrack }^{{x}_{n}} = {\mathrm{e}}^{a}. $$