# 式子复杂不要怕，先分析其“型”，再确定求解之“法”

## 一、题目

$$f(x) = \lim_{t \rightarrow x} \sin x \cdot \left( \frac{t}{x} \right)^{\frac{t^{3}}{t – x}}$$

$$\lim_{x \rightarrow 0} \frac{f(x) – x}{x^{3}} = ?$$

## 二、解析

\begin{aligned} f(x) & = \lim_{t \rightarrow x} \sin x \cdot \left( \frac{t}{x} \right)^{\frac{t^{3}}{t – x}} \\ \\ & = \sin x \cdot \lim_{t \rightarrow x} \left( \frac{t}{x} \right)^{\frac{t^{3}}{t – x}} \end{aligned}

\begin{aligned} f(x) & = \lim_{t \rightarrow x} \sin x \cdot \left( \frac{t}{x} \right)^{\frac{t^{3}}{t – x}} \\ \\ & = \sin x \cdot \lim_{t \rightarrow x} \left( \frac{t}{x} \right)^{\frac{t^{3}}{t – x}} \\ \\ & = \sin x \cdot \lim_{t \rightarrow x} \left( 1 + \frac{t}{x} – 1 \right)^{\frac{t^{3}}{t – x}} \\ \\ & = \sin x \cdot \lim_{t \rightarrow x} \left( 1 + \frac{t – x}{x} \right)^{\frac{t^{3}}{t – x}} \\ \\ & = \sin x \cdot \lim_{t \rightarrow x} \left( 1 + \frac{t – x}{x} \right)^{\frac{x}{t – x} \cdot \frac{t^{3}}{t – x} \cdot \frac{t – x}{x}} \\ \\ & = \sin x \cdot \lim_{x \rightarrow x} e^{\frac{t^{3}}{x}} \\ \\ & = \sin x \cdot e^{x^{2}} \end{aligned}

\begin{aligned} \lim_{x \rightarrow 0} \frac{f(x) – x}{x^{3}} & = \lim_{x \rightarrow 0} \frac{\sin x \cdot e^{x^{2}} – x}{x^{3}} \\ \\ & = \lim_{x \rightarrow 0} \frac{\sin x \cdot e^{x^{2}} – \sin x + \sin x – x}{x^{3}} \\ \\ & = \lim_{x \rightarrow 0} \frac{\sin x \cdot e^{x^{2}} – \sin x}{x^{3}} + \lim_{x \rightarrow 0} \frac{\sin x – x}{x^{3}} \\ \\ & = \lim_{x \rightarrow 0} \frac{\sin x \cdot (e^{x^{2}} – 1)}{x^{3}} + \lim_{x \rightarrow 0} \frac{ – \frac{1}{6} x^{3} }{x^{3}} \\ \\ & = 1 – \frac{1}{6} = \frac{5}{6} \end{aligned}