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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{2x}-\text{e}^\text{x}}{\sin2\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{2x}-\text{e}^\text{x}}{\sin2\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{2x}-\text{1}}{\sin2\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{x}-\text{1}}{\sin2\text{x}}$
$=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{2x}-\text{1}}{2\text{x}}\times\lim\limits_{\text{x}\rightarrow0}\frac{\text{2x}}{\sin2\text{x}}\Big)-\frac{1}{2}\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{x}-1}{\text{x}}\times\lim\limits_{\text{x}\rightarrow0}\frac{\text{2x}}{\sin2\text{x}}\Big)$
$=1-\frac{1}{2}$
$=\frac12$

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