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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{cosec x}-\cot\text{x}}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}\Big)\times\frac{1}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{1}{\sin\text{x}}\Big(\frac{1-\cos\text{x}}{\text{x}}\Big)\Big)$
$=\lim\limits_{\text{x}\rightarrow0}\bigg(\frac{1}{\sin\text{x}}\bigg(\frac{2\sin^2\frac{\text{x}}{2}}{\text{x}}\bigg)\bigg)$
$=2\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{1}{\frac{\sin\text{x}}{\text{x}}}\times\text{x}\bigg(\frac{\sin\frac{\text{x}}{2}}{\frac{\text{x}}{2}}\bigg)\times\frac{\text{x}}{4}\Bigg)$
$=2\bigg(\lim\limits_{\text{x}\rightarrow0}\frac{1}{\frac{\sin\text{x}}{\text{x}}}\bigg)\times\frac{1}{\text{x}}\times\bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\frac{\text{x}}{2}}{\frac{\text{x}}{2}}\bigg)\times\frac{\text{x}}{4}$
$=2\times\frac{1}{\text{x}}\times\frac{\text{x}}{4}$
$=\frac{1}{2}$

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