The value of lim _{x rightarrow 0} operatorna | vedclass

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(D) Let $L = \lim _{x \rightarrow 0} \frac{\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}}{\sin x}$.Rationalizing the numerator,we get:$L = \li

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$-\frac{1}{2 \sqrt{5}}$

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(D) Let $L = \lim _{x \rightarrow 0} \frac{\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}}{\sin x}$.Rationalizing the numerator,we get:$L = \lim _{x}$ ${\rightarrow 0} \frac{(2 \cos ^2 x+3 \cos x) - (\cos ^2 x+\sin x+4)}{\sin x (\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4})}$$L = \lim _{x}$ ${\rightarrow 0} \frac{\cos ^2 x+3 \cos x - \sin x - 4}{\sin x (\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4})}$Since $\cos ^2 x + 3 \cos x - 4 = (\cos x - 1)(\cos x + 4)$,we have:$L = \lim _{x}$ ${\rightarrow 0} \frac{(\cos x - 1)(\cos x + 4) - \sin x}{\sin x (\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4})}$Using $\cos x - 1 = -2 \sin ^2 \frac{x}{2}$ and $\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$:$L = \lim _{x}$ ${\rightarrow 0} \frac{-2 \sin ^2 \frac{x}{2}(\cos x + 4) - 2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2} (\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4})}$Dividing numerator and denominator by $2 \sin \frac{x}{2}$:$L = \lim _{x}$ ${\rightarrow 0} \frac{-\sin \frac{x}{2}(\cos x + 4) - \cos \frac{x}{2}}{\cos \frac{x}{2} (\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4})}$As $x \rightarrow 0$,$\sin \frac{x}{2} \rightarrow 0$,$\cos \frac{x}{2} \rightarrow 1$,$\cos x \rightarrow 1$:$L = \frac{-(0)(5) - 1}{1 (\sqrt{2+3} + \sqrt{1+0+4})} = \frac{-1}{\sqrt{5} + \sqrt{5}} = -\frac{1}{2 \sqrt{5}}$.

The value of $\lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)$ is

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