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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\tan\text{x}}{1-\cos\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\sin\text{x}}{\cos\text{x}(1-\cos\text{x})}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\sin\text{x}}{\cos\text{x}\big(2\sin^2\frac{\text{x}}{2}\big)}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)}{\cos\text{x}\big(2\sin^2\frac{\text{x}}{2}\big)}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\cos\frac{\text{x}}{2}}{\cos\text{x}\frac{\sin{\text{x}}{}}{2}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\cos\frac{\text{x}}{2}}{\cos\text{x}\frac{\sin\text{x}}{2}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1}{\cos\text{x}\times\frac{\frac{\tan\text{x}}{2}}{\text{x}}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1}{\cos\text{x}}\times\frac{1}{\cos\text{x}\times\frac{\frac{\tan\text{x}}{2}}{\frac{\text{x}}{2}}\times\frac12}$
$=1\times1\times2$
$=2$

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