求极限例题#1

\[求\lim_{x \to 0} \frac{tan2x}{x} \]

分子分母均乘以2: $$ \lim_{x \to 0} \frac{2tan2x}{2x} $$
$\because tanx=\frac{sinx}{cosx} $

\[ \therefore \lim_{x \to 0}\frac{2tan2x}{2x}= \lim_{x \to 0}\frac{2sin2x}{2x} \cdot \lim_{x \to 0} \frac{1}{cos2x} \Rightarrow 2\lim_{x \to 0}\frac{sin2x}{2x} \cdot \lim_{x \to 0} \frac{1}{cos2x} \]

$\because x \to 0 ==2x\to 0, 且cosx(x\to 0)=1 $

\[\therefore \lim_{x \to 0} \frac{1}{cos2x} =1 \]

\[\because \lim_{x \to 0} \frac{sinx}{x} =1 (第一个重要极限) \]

\[\therefore \lim_{x \to 0} \frac{sin2x}{2x} =1 \]

最终:$\quad 2\cdot 1\cdot 1=2 $

\[\lim_{x \to 0} \frac{tan2x}{x}=2 \]