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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^2\tan2\text{x}}{\tan\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^2\tan2\text{x}}{\tan\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\big(\frac{\text{x}^2}{2\text{x}}-\frac{\tan2\text{x}}{2\text{x}}\big)\times2\text{x}}{\frac{\tan\text{x}}{\text{x}}\times\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\big(\frac{\text{x}}{2}-\frac{\tan2\text{x}}{2\text{x}}\big)}{\frac{\tan\text{x}}{\text{x}}}$
$=2\Big(\frac{0-1}{1}\Big)$
$=-2$

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