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