# 微分方程和洛必达运算的结合

## 二、解析

### 求解过程

$$\textcolor{springgreen}{\lim \limits_{x \rightarrow 0} y(x)} = y(0) = \textcolor{springgreen}{0}$$

$$\textcolor{springgreen}{\lim \limits_{x \rightarrow 0} y^{\prime}(0)} = y^{\prime}(0) = \textcolor{springgreen}{0}$$

$$y^{\prime \prime} + 2 y^{\prime} + y = \mathrm{e}^{3 x} \Rightarrow$$

$$y^{\prime \prime} = \mathrm{e}^{3 x} – 2 y^{\prime} – y$$

\begin{aligned} \textcolor{springgreen}{\lim \limits_{x \rightarrow 0} y^{\prime \prime}(x)} \\ \\ & = \lim \limits_{x \rightarrow 0}\left[\mathrm{e}^{3x} – 2 y^{\prime}(x)-y(x)\right] \\ \\ & = 1- 0 – 0 \\ \\ & = \textcolor{springgreen}{1} \end{aligned}

\begin{aligned} \lim \limits_{x \rightarrow 0} \frac{\sin x^{2}}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{x^{2}}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{2x}{y^{\prime}(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{2}{y^{\prime \prime}(x)} \\ \\ & = \frac{2}{1} = \textcolor{orangered}{2 \neq 1} \end{aligned}

\begin{aligned} \lim \limits_{x \rightarrow 0} \frac{\sin x}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{x}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{1}{y^{\prime}(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{0}{y^{\prime \prime}(x)} \\ \\ & = \frac{0}{1} = \textcolor{orangered}{0 \neq 1} \end{aligned}

\begin{aligned} \lim \limits_{x \rightarrow 0} \frac{\ln \sqrt{1+x^{2}}}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{\frac{1}{2} \ln \left(1+x^{2}\right)}{y(x)} \\ \\ & = \frac{1}{2} \lim \limits_{x \rightarrow 0} \frac{x^{2}}{y(x)} \\ \\ & = \frac{1}{2} \lim \limits_{x \rightarrow 0} \frac{2 x}{y^{\prime}(x)} \\ \\ & = \frac{1}{2} \lim \limits_{x \rightarrow 0} \frac{2}{y^{\prime \prime}(x)} \\ \\ & = \textcolor{springgreen}{\boldsymbol{\frac{1}{2} \times 2 = 1}} \end{aligned}

\begin{aligned} \lim \limits_{x \rightarrow 0} \frac{\ln \left(1+x^{2}\right)}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{x^{2}}{y(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{2 x}{y^{\prime}(x)} \\ \\ & = \lim \limits_{x \rightarrow 0} \frac{2}{y^{\prime \prime}(x)} \\ \\ & = \textcolor{orangered}{2 \neq 1} \end{aligned}