没说邻域内可导不能用洛必达法则

一、题目

已知，函数 $f(x)$ 在点 $x_{0}$ 处可导, $\left\{ \alpha_{n} \right\}$ 与 $\left\{\beta_{n} \right\}$ 是两个趋于 0 的正数列, 请求解下面的极限：

$$
I=\lim _{n \rightarrow \infty} \frac{f \left(x_{0} + \alpha_{n} \right) – f \left(x_{0} – \beta_{n} \right)}{\alpha_{n} + \beta_{n} }
$$

难度评级：

二、解析

Tip

本题只说了函数 $f(x)$ 在点 $x_{0}$ 这一点处可导，没有说函数 $f(x)$ 在点 $x_{0}$ 的邻域内是否可导，此时我们只能将函数 $f(x)$ 看作在点 $x_{0}$ 的邻域内不可导，因此不能使用洛必达法则，只能使用一点处导数的定义式。

zhaokaifeng.com

简单解法

Note

在简单解法中，我们不考虑无穷小对极限运算的影响。

zhaokaifeng.com

根据函数在一点处导数的定义式，可知：

$$
\begin{aligned}
& \frac{f \left(x_{0} + \alpha_{n} \right) – f \left(x_{0} \right)}{\alpha_{n}} = f^{\prime}\left(x_{0} \right) \\ \\
& \frac{f \left(x_{0} – \beta_{n} \right) – f \left(x_{0} \right)}{-\beta_{n}} = f^{\prime}\left( x_{0} \right)
\end{aligned}
$$

于是：

$$
\begin{aligned}
I & =\lim _{n \rightarrow \infty} \left[\frac{f \left(x_{0} + \alpha_{n} \right) – f\left( x_{0} \right)}{\alpha_{n}} \frac{\alpha_{n}}{\alpha_{n} + \beta_{n} } + \frac{f \left(x_{0} – \beta_{n} \right) – f\left( x_{0} \right)}{-\beta_{n} } \frac{\beta_{n}}{\alpha_{n} + \beta_{n} }\right] \\ \\
& =\lim _{n \rightarrow \infty}\left[f^{\prime}\left( x_{0} \right) \frac{\alpha_{n} }{\alpha_{n} + \beta_{n}} + f^{\prime} \left( x_{0} \right) \frac{\beta_{n} }{\alpha_{n} + \beta_{n} } \right] \\ \\
& =\lim _{n \rightarrow \infty}\left[f^{\prime}\left( x_{0} \right) \cdot \frac{\alpha_{n} + \beta_{n} }{\alpha_{n} + \beta_{n}} \right] \\ \\
& =\lim _{n \rightarrow \infty}\left[f^{\prime}\left( x_{0} \right) \cdot 1 \right] \\ \\
& = \textcolor{springgreen}{\boldsymbol{ f^{\prime}\left( x_{0} \right) }}
\end{aligned}
$$

严格解法

Note

在严格解法中，我们需要考虑无穷小对极限运算的影响。

zhaokaifeng.com

根据“扩展的一点处导数定义式”，可知：

$$
\begin{aligned}
& \frac{f \left(x_{0} + \alpha_{n} \right) – f \left( x_{0} \right)}{\alpha_{n} } = f^{\prime} \left(x_{0} \right) + \textcolor{orange}{z_{n}} \\ \\
& \frac{f \left(x_{0} – \beta_{n} \right) – f \left( x_{0} \right)}{-\beta_{n} } = f^{\prime}\left( x_{0} \right) + \textcolor{orange}{k_{n}}
\end{aligned}
$$

其中 $\textcolor{orange}{z_{n}}$ 与 $\textcolor{orange}{k_{n}}$ 都是 $n \rightarrow \infty$ 时的无穷小量。

于是：

$$
\begin{aligned}
I & = \lim _{n \rightarrow \infty}\left[\frac{f \left(x_{0} + \alpha_{n} \right) – f \left( x_{0} \right)}{\alpha_{n}} \frac{\alpha_{n}}{\alpha_{n} + \beta_{n}} + \frac{f \left(x_{0} – \beta_{n} \right) – f \left( x_{0} \right)}{-\beta_{n}} \frac{\beta_{n}}{\alpha_{n} + \beta_{n} }\right] \\ \\
& = \lim _{n \rightarrow \infty}\left[\left(f^{\prime} \left( x_{0} \right) + \textcolor{orange}{z_{n}} \right) \frac{\alpha_{n}}{\alpha_{n} + \beta_{n}} + \left(f^{\prime} \left(x_{0} \right) + \textcolor{orange}{k_{n}} \right) \frac{\beta_{n} }{\alpha_{n} + \beta_{n} } \right] \\ \\
& = \lim _{n \rightarrow \infty}\left[f^{\prime}\left( x_{0} \right) + \frac{\alpha_{n} \textcolor{orange}{z_{n}} + \beta_{n} \textcolor{orange}{k_{n}} }{\alpha_{n} + \beta_{n} } \right] \\ \\
& = f^{\prime}\left( x_{0} \right) + \textcolor{orangered}{ \lim _{n \rightarrow \infty} \frac{\alpha_{n} \textcolor{orange}{z_{n}} + \beta_{n} \textcolor{orange}{k_{n}} }{\alpha_{n} + \beta_{n} } }
\end{aligned}
$$

又因为，当 $n \rightarrow \infty$ 的时候，有：

$$
\begin{aligned}
0 & \leqslant \left| \textcolor{orangered}{ \lim _{n \rightarrow \infty} \frac{\alpha_{n} \textcolor{orange}{z_{n}} + \beta_{n} \textcolor{orange}{k_{n}} }{\alpha_{n} + \beta_{n} } } \right| \\ \\
\Rightarrow \ & \leqslant \left| \lim_{x \rightarrow \infty} \left( \frac{\alpha_{n} \textcolor{orange}{ z_{n} }}{\alpha_{n} + \beta_{n}} + \frac{\beta_{n} \textcolor{orange}{ k_{n} }}{\alpha_{n} + \beta_{n}} \right) \right| \\ \\
\Rightarrow \ & \textcolor{gray}{
\begin{cases}
\frac{\alpha_{n}}{\alpha_{n} + \beta_{n}} < 1 \\ \\
\frac{\beta_{n}}{\alpha_{n} + \beta_{n}} < 1
\end{cases}
} \\ \\
\Rightarrow \ & \leqslant \left| \lim_{n \rightarrow \infty} \left( z_{n} + k_{n} \right) \right| \\ \\
\Rightarrow \ & \leqslant \left| \lim_{n \rightarrow \infty} \textcolor{orange}{ z_{n} } \right| + \left| \lim_{n \rightarrow \infty} \textcolor{orange}{ k_{n} } \right| \rightarrow 0
\end{aligned}
$$

即：

$$
\left| \textcolor{orangered}{ \lim _{n \rightarrow \infty} \frac{\alpha_{n} \textcolor{orange}{z_{n}} + \beta_{n} \textcolor{orange}{k_{n}} }{\alpha_{n} + \beta_{n} } } \right| \rightarrow 0 ^{+}
$$

综上可知：

$$
\textcolor{springgreen}{
\boldsymbol{
I = f^{\prime} \left( x_{0} \right)
}
}
$$

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