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

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow 0}\frac{1-\cos2\text{x}}{\text{x}^{2}}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow 0}\frac{1-\cos2\text{x}}{\text{x}^{2}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin^{2}\text{x}}{\text{x}^{2}}$
$=\lim\limits_{\text{x} \rightarrow 0}2\big(\frac{\sin\text{x}}{\text{x}}\big)^{2}$
$=2\times1=2$
Hence, the required answer is 2.

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