Evaluate the following limit: _ x 0 e^ 3+x- -e^3 x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{ 3+x}-\sin\text{x}-\text{e}^3}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{ 3+x}-\sin\text{x}-\text{e}^3}{\text{x}}$
$=\text{e}^3\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{ x}-1}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}$
$=\text{e}^3 \text{log e}-1$
$=\text{e}^3 -1$

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