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

Question

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

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Answer

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

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