Evaluate: _ x 0 1- 2 x x^2 . - Vidyadip

Question

Evaluate: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$.

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Answer

We have to find the value of
$\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$
$=\lim _{x \rightarrow 0} \frac{2 \sin ^2 x}{x^2}$
$=2 \lim _{x \rightarrow 0}\left(\frac{\sin x}{x} \times \frac{\sin x}{x}\right)$
$=2 \lim _{x \rightarrow 0} \frac{\sin x}{x} \times \lim _{x \rightarrow 0} \frac{\sin x}{x}$
$=2(1)(1)$
$=2$

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