极限例题(一)
First
\[\begin{align}
\lim_{x \to 0} \frac{1-\cos{x}}{x^{2} } =?
\\ \\
\because 1-\cos{x} = \sin^{2}{x}
\\
\therefore \lim_{x \to 0} \frac{1-\cos{x}}{x^{2} } \Rightarrow \lim_{x \to 0} \frac{\sin^{2}{x}}{x^{2} }
\\ \\
分子分母同除2: \enspace \lim_{x \to 0} \frac{ \sin^{2} \frac{x}{2} }{\frac{x^2}{2} }
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分母变形: \enspace \lim_{x \to 0} \frac{ \sin^{2}\frac{x}{2} }{2\cdot \frac{x^2}{4} }
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\Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} \frac{ \sin^{2} \frac{x}{2} }{\frac{x^2}{4} }
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\Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} ( \frac{\sin\frac{x}{2}} {\frac{x}{2}})^{2}
\\ \\
根据第一重要极限:
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\therefore \frac{1}{2} \lim_{x \to 0} {1}^{2}=\frac{1}{2}
\end{align}
\]
