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

Question

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

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Answer

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

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