Evaluate the following limit: _ x 6 cosec^2 x-3 cosec x-2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}\frac{\text{cosec}^2\text{ x}-3}{\text{cosec}\text{ x}-2}$

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Answer

$\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}\frac{\text{cosec}^2\text{x}-3}{\text{cosec }\text{x}-2}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}\frac{\text{cosec}^2\text{x}-4}{\text{cosec }\text{x}-2}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}\frac{(\text{cosec }\text{x}-2)(\text{cosec x}+2)}{(\text{cosec }\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}(\text{cosec }\text{x}+2)$
$=\text{cosec}\frac{\pi}{6}+2$
$=2+2=4$

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