\(\lim _{x\to 0}\csc x\left(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}\right)\) is [JEE Main 2025, 24 Jan…

\(\lim _{x\to 0}\csc x\left(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}\right)\)

\(=\lim _{x\to 0}\frac{2{\cos }^{2}x+3\cos x-{\cos }^{2}x-\sin x-4}{\sin x\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\)

\(=\lim _{x\to 0}\frac{{\cos }^{2}x+3\cos x-\sin x-4}{\sin x(\sqrt{5}+\sqrt{5})}\)

on applying, L'Hospitals rule,

\(=\lim _{x\to 0}\frac{-2\cos x\sin x-3\sin x-\cos x}{\cos x\cdot 2\sqrt{5}}=-\frac{1}{2\sqrt{5}}\)