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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{\text{e}^\text{x}-1}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{\text{e}^\text{x}-1}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{\text{2x}}\times2\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{\text{e}^\text{x}-1}$
$=1\times2\times\text{log}\text{ e}$
$=2$

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