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Question
\[\lim_{x \to \frac{\pi}{6}} \frac{\cot^2 x - 3}{cosec x - 2}\]
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Solution
\[\lim_{x \to \frac{\pi}{6}} \left[ \frac{\cot^2 x - 3}{cosec x - 2} \right]\]
\[ = \lim_{x \to \frac{\pi}{6}} \left[ \frac{{cosec}^2 x - 1 - 3}{cosec x - 2} \right]\]
\[ = \lim_{x \to \frac{\pi}{6}} \left[ \frac{{cosec}^2 x - 4}{cosec x - 2} \right]\]
\[ = \lim_{x \to \frac{\pi}{6}} \left[ \frac{\left( cosec x - 2 \right) \left( cosec x + 2 \right)}{\left( cosec x - 2 \right)} \right]\]
\[ = cosec\frac{\pi}{6} + 2\]
\[ = 2 + 2\]
\[ = 4\]
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