Evaluate: _ x 6 ^ 2 x-3 cosecx-2 - Vidyadip

Question

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

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Answer

Given that $\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\cot^{2}\text{x}-3}{\text{cosec}\text{x}-2}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\text{cosec}^{2}\text{x}-1-3}{\text{cosec}\text{x}-2}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\text{cosec}^{2}\text{x}-4}{\text{cosec}\text{x}-2}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{(\text{cosec}\text{x}+2)(\text{cosec}\text{x}-2)}{(\text{cosec}\text{x}-2)}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}({\text{cosec}\text{x}+2})$
Taking limit we have
$=\text{cosec}\frac{\pi}{6}+2=2+2=4$
Hence, the required answer is 4.

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