Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& \underset{x\to 0}{\lim }\mathrm{cosec}\left(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}\right)\\ & \underset{x\to 0}{\lim }\frac{\cos \mathrm{ec}x\left({\cos }^{2}x+3\cos x-\sin x-4\right)}{\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\\ & \underset{x\to 0}{\lim }\frac{1}{\sin x}\frac{\left({\cos }^{2}x+3\cos x-4\right)-\sin x}{\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\\ & \underset{x\to 0}{\lim }\frac{\left({\cos }^x+4\right)(\cos x-1)-\sin x}{\sin x\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\\ & \underset{x\to 0}{\lim }\frac{-2{\sin }^2\frac{x}{2}(\cos x+4)-2\sin \frac{x}{2}\cos \frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\\ & \underset{x\to 0}{\lim }\frac{-\left({\sin }^x(\cos x+4)+\cos \frac{x}{2}\right)}{\cos \frac{x}{2}\left(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4}\right)}\\ & -\frac{1}{2\sqrt{5}}\end{array}$