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Question
\[\lim_{x \to 0} \frac{cosec x - \cot x}{x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{cosec x - \cot x}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{1 - \cos x}{x \sin x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin^2 \left( \frac{x}{2} \right)}{x \sin x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin^2 \left( \frac{x}{2} \right)}{\left( \frac{x}{2} \right)^2} \times \frac{\frac{x^2}{4}}{\frac{x \sin x}{x \times x} \times x^2} \right]\]
\[ = 2 \times \frac{1}{4}\]
\[ = \frac{1}{2}\]
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