# 2017年考研数二第15题解析：变限积分、洛必达法则、无穷小

## 题目

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{x – t} e^{t} \mathrm{d} t}{\sqrt{x^{3}}}.$$

## 解析

### 方法一

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{x – t} e^{t} \mathrm{d} t}{\sqrt{x^{3}}} \Rightarrow$$

$$令 u = x – t \Rightarrow$$

$${\color{White} \left\{\begin{matrix} u = x – t;\\ t = x – u. \end{matrix}\right. \Rightarrow }$$

$${\color{White} \left\{\begin{matrix} t \in (0,x);\\ u \in (x , 0). \end{matrix}\right. \Rightarrow }$$

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{x}^{0} \sqrt{u} e^{x-u} \mathrm{d} (x-u)}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{ e^{x} \int_{x}^{0} \sqrt{u} e^{-u} [-\mathrm{d} (u)]}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{u} e^{-u} \mathrm{d} u}{\sqrt{x^{3}}} \Rightarrow$$

[1]. $\lim_{x \rightarrow 0^{+}} e^{x} = 0$.

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{u} e^{-u} \mathrm{d} u}{x^{\frac{3}{2}}} \Rightarrow$$

$${\color{White}使用洛必达法则，分子分母同时求导 \Rightarrow}$$

$$\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x} e^{-x}}{\frac{3}{2}x^{\frac{1}{2}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x}}{\frac{3}{2}\sqrt{x}} = \frac{2}{3}.$$

### 方法二

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{x – t} e^{t} \mathrm{d} t}{\sqrt{x^{3}}} \Rightarrow$$

$$令 u = \sqrt{x – t} \Rightarrow$$

$${\color{White} \left\{\begin{matrix} u = \sqrt{x – t};\\ t = x – u^{2}. \end{matrix}\right. \Rightarrow }$$

$${\color{White} \left\{\begin{matrix} t \in (0, x);\\ u \in (\sqrt{x}, 0). \end{matrix}\right. \Rightarrow }$$

$$\lim_{x \rightarrow 0^{+}} \frac{\int_{\sqrt{x}}^{0} u e^{x-u^{2}} \mathrm{d} (x – u^{2})}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{e^{x} \int_{\sqrt{x}}^{0} u e^{-u^{2}} [-2u \mathrm{d} (u)]}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{e^{x} \int_{0}^{\sqrt{x}} u e^{-u^{2}} [2u \mathrm{d} (u)]}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{2 e^{x} \int_{0}^{\sqrt{x}} u^{2} e^{-u^{2}} \mathrm{d} u}{\sqrt{x^{3}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{2 e^{x} \int_{0}^{\sqrt{x}} u^{2} e^{-u^{2}} \mathrm{d} u}{x^{\frac{3}{2}}} \Rightarrow$$

$${\color{White}使用洛必达法则，分子分母同时求导 \Rightarrow}$$

$$\lim_{x \rightarrow 0^{+}} \frac{2 e^{x} x e^{-x} (\sqrt{x})^{‘}}{\frac{3}{2} x^{\frac{1}{2}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{2 x (\frac{1}{2} x ^{\frac{-1}{2}})}{\frac{3}{2} x^{\frac{1}{2}}} \Rightarrow$$

$$\lim_{x \rightarrow 0^{+}} \frac{x^{\frac{1}{2}}}{\frac{3}{2} x^{\frac{1}{2}}} = \frac{2}{3}.$$