lim _{x rightarrow 0} frac{(operatorname{cose | vedclass

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(D) We have $\operatorname{cosec} x - \cot x = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1-\cos x}{\sin x} = 	an(x/2)$.Also,$e^x - e^{-x} = 2

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$4 \sqrt{3}$

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(D) We have $\operatorname{cosec} x - \cot x = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1-\cos x}{\sin x} = 	an(x/2)$.Also,$e^x - e^{-x} = 2 \sinh x \approx 2x$ as $x 	o 0$.So,the numerator is $	an(x/2) \cdot (e^x - e^{-x}) \approx (x/2) \cdot (2x) = x^2$.For the denominator,$\sqrt{3} - \sqrt{2+\cos x} = \frac{3 - (2+\cos x)}{\sqrt{3} + \sqrt{2+\cos x}} = \frac{1-\cos x}{\sqrt{3} + \sqrt{2+\cos x}}$.Using $1-\cos x \approx x^2/2$,the denominator is $\frac{x^2/2}{\sqrt{3} + \sqrt{3}} = \frac{x^2}{4\sqrt{3}}$.Thus,the limit is $\lim_{x 	o 0} \frac{x^2}{x^2 / (4\sqrt{3})} = 4\sqrt{3}$.

$\lim _{x \rightarrow 0} \frac{(\operatorname{cosec} x-\cot x)(e^x-e^{-x})}{\sqrt{3}-\sqrt{2+\cos x}} = $

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