Evaluate the following limit: _ x 0 1- 2x 3 ^2x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos2\text{x}}{3\tan^2\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos2\text{x}}{3\tan^2\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin^2\text{x}}{3\tan^2\text{x}}$
$=\frac23\lim\limits_{\text{x} \rightarrow0}\frac{\sin^2\text{x}}{\Big(\frac{\sin^2\text{x}}{\cos^2\text{x}}\Big)}$
$=\frac23\lim\limits_{\text{x} \rightarrow0}\cos^2\text{x}$
$=\frac23$

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