极限例题(一)

First

$$\begin{align} \lim_{x \to 0} \frac{1-\cos{x}}{x^{2} } =? \\ \\ \because 1-\cos{x} = \sin^{2}{x} \\ \therefore \lim_{x \to 0} \frac{1-\cos{x}}{x^{2} } \Rightarrow \lim_{x \to 0} \frac{\sin^{2}{x}}{x^{2} } \\ \\ 分子分母同除2: \enspace \lim_{x \to 0} \frac{ \sin^{2} \frac{x}{2} }{\frac{x^2}{2} } \\ \\ 分母变形: \enspace \lim_{x \to 0} \frac{ \sin^{2}\frac{x}{2} }{2\cdot \frac{x^2}{4} } \\ \\ \Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} \frac{ \sin^{2} \frac{x}{2} }{\frac{x^2}{4} } \\ \\ \Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} ( \frac{\sin\frac{x}{2}} {\frac{x}{2}})^{2} \\ \\ 根据第一重要极限: \\ \\ \therefore \frac{1}{2} \lim_{x \to 0} {1}^{2}=\frac{1}{2} \end{align}$$

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