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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{ x }-\text{e }^{\sin\text{x}}}{\text{x}-\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0 }\text{e}^{ \sin \text{x}}\Big[\frac{\text{e}^{\text{ x }-\sin\text{x}}-1}{\text{x}-\sin\text{x}}\Big]$
$=1\times\text{log e}$
$=1$

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