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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\cot^2\text{x}-3}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-1\big)-3}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-4\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec }\text{x}-2\big)\big(\text{cosec }\text{x}+2\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\text{cosec }\text{x}+2$
$=\text{cosec}\frac{\pi}{6}+2$
$=2+2$
$=4$

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