通过提取式子中的“大头”，变无穷大为无穷小

一、题目

$$
I = \lim \limits_{x \rightarrow \infty} \frac{(2 x-3)^{20}(3 x+2)^{30}}{(2 x+1)^{50}+x^{48}(2 x-1)} = ?
$$

难度评级：

二、解析

$$
\begin{aligned}
I & =\lim \limits_{x \rightarrow \infty} \frac{\frac{1}{x^{50}}(2 x-3)^{20}(3 x+2)^{30}}{\frac{1}{x^{50}}\left[(2 x+1)^{50}+x^{48}(2 x-1)\right]} \\ \\
& =\lim \limits_{x \rightarrow \infty} \frac{\frac{1}{x^{20}}(2 x-3)^{20} \cdot \frac{1}{x^{30}} \cdot (3 x+2)^{30}}{ \frac{1}{x^{50}} \left[(2 x+1)^{50}+x^{48}(2 x-1)\right]} \\ \\
& =\lim \limits_{x \rightarrow \infty} \frac{\left(2-\frac{3}{x}\right)^{20} \cdot （3 + \frac{2}{x})^{30}}{\left(2+\frac{1}{x}\right)^{50}+\frac{2 x-1}{x^{2}}} \\ \\
& =\lim \limits_{x \rightarrow \infty} \frac{2^{20} \cdot 3^{30}}{2^{50}+0}=\lim \limits_{x \rightarrow \infty} \frac{3^{30}}{2^{30}}=\left(\frac{3}{2}\right)^{30}
\end{aligned}
$$

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