Evaluate the following limit: _ x 0 (cosec x- ) - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}(\text{cosec x}-\cot\text{x})$

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Answer

$\lim\limits_{\text{x}\rightarrow0}(\text{cosec x}-\cot\text{x})$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}\Big)$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin^2\frac{\text{x}}{2}}{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{2}$
$=\bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\frac{\tan\text{x}}{2}}{\frac{\text{x}}{2}}\bigg)\times\frac{\text{x}}{2}$
$=\lim\limits_{\text{x}\rightarrow0}1\times\frac{\text{x}}{2}$
$=0$

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