用 e 抬起法去根号

一、题目

$$I=\underset{n\to \mathrm{\infty }}{lim}\frac{n}{\ln (n+1)}(\sqrt[n]{n}–1)$$

难度评级：

二、解析

已知：

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\ln n}{n}\Rightarrow \text{洛必达}\Rightarrow \underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n}}{1}=0$$

于是：

$$\begin{array}{rl}I=& \underset{n\to \mathrm{\infty }}{lim}\frac{n}{\ln (n+1)}(\sqrt[n]{n}–1)\\ =& \underset{n\to \mathrm{\infty }}{lim}\frac{n}{\ln (n+1)}(e^{\frac{1}{n}\ln n}–1)\\ =& \underset{n\to \mathrm{\infty }}{lim}\frac{n}{\ln (n+1)}(\frac{1}{n}\ln n)\\ =& \underset{n\to \mathrm{\infty }}{lim}\frac{\ln n}{\ln (n+1)}\\ =& \underset{n\to \mathrm{\infty }}{lim}\frac{\ln n}{\ln n}=1\end{array}$$

---