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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{x}-1}{\sqrt{1-\cos\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{e}^\text{x}-1\big)\big(\sqrt{1+\cos\text{x}}\big)}{\big(\sqrt{1-\cos\text{x}}\big)\big(\sqrt{1+\cos\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{e}^\text{x}-1\big)\big(\sqrt{1+\cos\text{x}}\big)}{\sin\text{x}}$
Both numerator and denominator are both zeros for x = 0 hence limit can not exist.

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