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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\big(\text{e}^\text{x}-1\big)}{1-\cos\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\big(\text{e}^\text{x}-1\big)}{2\sin^2\big(\frac{\text{x}}{2}\big)}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{e}^\text{x}-1\big)}{2\text{x}}\times\lim\limits_{\text{x}\rightarrow0}\frac{4}{\begin{pmatrix}\frac{\sin\big(\frac{\text{x}}{2}\big)}{\frac{\text{x}}{2}}\end{pmatrix}^2}$ $=\frac{1}{2}\times4$ $=2$

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