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