极限证题例一
作者：互联网
\[求\lim_{x \to 0} \frac{1-cosx}{x^{2} }
\]\[\because 1-cosx = sin^{2}x
\]\[\therefore \lim_{x \to 0} \frac{1-cosx}{x^{2} }
\]\[\Rightarrow \lim_{x \to 0} \frac{sin^{2}x}{x^{2} }
\]\[分子分母同时除以2 \Rightarrow \lim_{x \to 0} \frac{ sin^{2} \frac{x}{2} }{\frac{x^2}{2} }
\]\[分母变形 \Rightarrow \lim_{x \to 0} \frac{ sin^{2} \frac{x}{2} }{2\cdot \frac{x^2}{4} }
\]\[\Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} \frac{ sin^{2} \frac{x}{2} }{\frac{x^2}{4} }
\]\[\Rightarrow \frac{1}{2} \cdot \lim_{x \to 0} ( \frac{sin \frac{x}{2}} {\frac{x}{2}})^{2}
\]\[\because sin\frac{x}{x}(x\to 0)=1
\]\[\therefore \frac{1}{2} \lim_{x \to 0} {1}^{2}=\frac{1}{2}
\]
标签：例一,frac,cdot,lim,极限,证题,cosx,sin,Rightarrow 来源： https://www.cnblogs.com/Preparing/p/16486064.html