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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\text{e}^{-\cos\text{x}}-1}{-\cos\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\text{e}^{-\cos\text{x}}-1}{-\cos\text{x}}$
Let $\text{ x}-\frac{\pi}{2}=\text{h}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^{-\sin\text{h}}-1}{-\sin\text{h}}$
$=\lim\limits_{\sin\text{h}\rightarrow0}\frac{\text{e}^{-\sin\text{h}}-1}{-\sin\text{h}}$
$=1$

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