Correct option is (4) \(-\frac{1}{2 \sqrt{5}}\)
\(\lim _{x \rightarrow 0} \operatorname{cosec x}\left(\sqrt{2 \cos ^{2} x+3 \cos x}-\sqrt{\cos ^{2} x+\sin x+4}\right)\)
\(\lim _{x \rightarrow 0} \frac{\operatorname{cosec} x\left(\cos ^{2} x+3 \cos x-\sin x-4\right)}{\left(\sqrt{2 \cos ^{2} x+3 \cos x}+\sqrt{\cos ^{2} x+\sin x+4}\right)}\)
\(\lim _{x \rightarrow 0} \frac{1}{\sin x} \frac{\left(\cos ^{2} x+3 \cos x-4\right)-\sin x}{\left(\sqrt{2 \cos ^{2} x+3 \cos x}+\sqrt{\cos ^{2} x+\sin x+4}\right)}\)
\(\lim _{x \rightarrow 0} \frac{(\cos x+4)(\cos x-1)-\sin x}{\sin x\left(\sqrt{2 \cos ^{2} x+3 \cos x}+\sqrt{\cos ^{2} x+\sin x+4}\right)}\)
\(\lim _{x \rightarrow 0} \frac{-2 \sin ^{2} \frac{x}{2}(\cos x+4)-2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}\left(\sqrt{2 \cos ^{2} x+3 \cos x}+\sqrt{\cos ^{2} x+\sin x+4}\right)}\)
\(\lim _{x \rightarrow 0} \frac{-\left(\sin \frac{x}{2}(\cos x+4)+\cos \frac{x}{2}\right)}{\cos \frac{x}{2}\left(\sqrt{2 \cos ^{2} x+3 \cos x}+\sqrt{\cos ^{2} x+\sin x+4}\right)}\)
\(-\frac{1}{2 \sqrt{5}}\)
