Evaluate the following limit: _ x 4 2-cosec^2x 1- - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{2-\text{cosec}^2\text{x}}{1-\cot\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{2-\text{cosec}^2\text{x}}{1-\cot\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{2-\big(1+\cot^2\text{x}\big)}{1-\cot\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{2-1-\cot^2\text{x}}{1-\cot\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{1-\cot^2\text{x}}{1-\cot\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{(1-\cot\text{x})(1+\cot\text{x})}{(1-\cot\text{x})}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}(1+\cot\text{x})$
$=1+\cot\frac{\pi}{4}$
$=1+1$
$=2$

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