# 怎么表示切线在 X 轴上的截距？

## 二、解析

$$x_{0} \neq 0 \Rightarrow\left(x_{0}, f\left(x_{0}\right)\right) \Rightarrow$$

$$y-f\left(x_{0}\right)=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) \Rightarrow$$

$$y=0 \Rightarrow$$

$$-f\left(x_{0}\right)=x f^{\prime}\left(x_{0}\right)-x_{0} f^{\prime}\left(x_{0}\right) \Rightarrow$$

$$x_{0}=u\left(x_{0}\right)=\frac{x_{0} f^{\prime}\left(x_{0}\right)-f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)} \Rightarrow$$

$$\textcolor{springgreen}{ \lim \limits_{x_{0} \rightarrow 0} \frac{x}{u(x)}=\frac{x f^{\prime}\left(x_{0}\right)}{x_{0} f^{\prime}\left(x_{0}\right)-f\left(x_{0}\right)} } \tag{1}$$

### 方法 1：洛必达

$$\lim \limits_{x \rightarrow 0} \frac{x}{u(x)}=\frac{x f^{\prime}(x)}{x f^{\prime}(x)-f(x)} \Rightarrow$$

$$\frac{0}{0} \Rightarrow \text{ 洛必达运算 } \Rightarrow$$

$$\frac{f^{\prime}\left(x_{0}\right)+x f^{\prime \prime}(x)}{f^{\prime}(x)+x f^{\prime \prime}(x)-f^{\prime}(x)}=\frac{x f^{\prime \prime}(x)+f^{\prime}(x)}{x f^{\prime \prime}(x)} =$$

$$1+\frac{f^{\prime}(x)}{x f^{\prime \prime}(x)}=1+\frac{f^{\prime}(x)-f^{\prime}(0)}{x-0} \cdot \frac{1}{f^{\prime \prime}(x)}=$$

$$1+\frac{f^{\prime \prime}(x)}{f^{\prime \prime}(x)}=1+1=2$$

### 方法 2：泰勒公式

$$\lim \limits_{x \rightarrow 0} \frac{x}{u(x)}=\frac{x f^{\prime}(x)}{x f^{\prime}(x)-f(x)} \Rightarrow$$

$$x f^{\prime}(x)=x\left[f^{\prime}(0)+f^{\prime \prime}(0) \cdot x+o(x)\right]=$$

$$x^{2} f^{\prime \prime}(0)+o\left(x^{2}\right)$$

$$f(x)=f(0)+f^{\prime}(0) \cdot x+\frac{1}{2} f^{\prime \prime}(0) \cdot x^{2}+0\left(x^{2}\right) =$$

$$\frac{1}{2} x^{2} f^{\prime \prime}(0)+0\left(x^{2}\right)$$

$$\lim \limits_{x \rightarrow 0} \frac{x}{u(x)}=\frac{x^{2} f^{\prime \prime}(0)+o\left(x^{2}\right)}{x^{2} f^{\prime \prime}(0)-\frac{1}{2} x^{2} f^{\prime \prime}(0) + o\left(x^{2}\right)}=2$$