# 趋于“无穷大”就要考虑趋于“负无穷大”和趋于“正无穷大”两种情况

## 二、解析

### 当 $x \rightarrow – \infty$ 时

\begin{aligned} \lim \limits_{x \rightarrow − \infty}\frac{\sqrt[3]{2x^{3} + 3}}{\sqrt{3x^{2}−2}} \\ \\ & = \frac{\sqrt[3]{\textcolor{springgreen}{x^{3}} \left(2 + 3x^{−3}\right)}}{\sqrt{\textcolor{orangered}{x^{2}} \left(3−2x^{−2}\right)} } \\ \\ & = \lim \limits_{x \rightarrow − \infty}\frac{\textcolor{springgreen}{x} \sqrt[3]{2 + 3x^{−3}}}{\textcolor{orangered}{\left(−x\right)} \sqrt{3−2x^{−2}}} \\ \\ & = \lim _ { x \rightarrow – \infty } \frac { \sqrt [ 3 ] { 2 + 3 x ^ { – 3 } } } { – \sqrt { 3 – 2 x ^ { – 2 } } } \\ \\ & = \frac { – \sqrt [ 3 ] { 2 + 0 } } { \sqrt { 3 + 0 } } \\ \\ & = \textcolor{green}{\boldsymbol{\frac { -\sqrt [ 3 ] { 2 } } { \sqrt { 3 } } }} \end{aligned}

### 当 $x \rightarrow + \infty$ 时

\begin{aligned} \lim \limits_{x \rightarrow + \infty}\frac{\sqrt[3]{2x^{3} + 3}}{\sqrt{3x^{2}−2}} \\ \\ & = \lim \limits_{x\rightarrow + \infty} \frac{\sqrt[3]{\textcolor{springgreen}{x^{3}} \left(2 + 3x^{−3}\right)}}{\sqrt{\textcolor{springgreen}{x^{2}} \left(3−2x^{−2}\right)}} \\ \\ & = \lim \limits_{x\rightarrow + \infty}\frac{\textcolor{springgreen}{x} \sqrt[3]{2 + 3x^{−3}}}{\textcolor{springgreen}{x} \sqrt{3−2x^{−2}}} \\ \\ & = \frac{\sqrt[3]{2 + 0}}{\sqrt{3 + 0}} \\ \\ & = \textcolor{green}{\boldsymbol{\frac{\sqrt[3]{2}}{\sqrt{3}}}} \end{aligned}

$$\lim _ { x \rightarrow \textcolor{orangered}{- \infty} } I \neq \lim _ { x \rightarrow \textcolor{springgreen}{+ \infty} } I$$