QJEEM25Ma736c: JEE Main 2025 Mathematics - Class 12 JEEM25Ma736c Solution: is:

$\begin{aligned}& \lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right) \\& \lim _{x \rightarrow 0} \frac{\operatorname{cosec}\left(\cos ^2 x+3 \cos x-\sin x-4\right)}{\left(\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4}\right)} \\& \lim _{x \rightarrow 0} \frac{1}{\sin x} \frac{\left(\cos ^2 x+3 \cos x-4\right)-\sin x}{\left(\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4}\right)} \\& \lim _{x \rightarrow 0} \frac{\left(\cos ^x+4\right)(\cos x-1)-\sin x}{\sin x\left(\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4}\right)} \\& \lim _{x \rightarrow 0} \frac{-2 \sin ^2 \frac{x}{2}(\cos x+4)-2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}\left(\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4}\right)} \\& \lim _{x \rightarrow 0} \frac{-\left(\sin ^x \frac{x}{2}(\cos x+4)+\cos \frac{x}{2}\right)}{\cos \frac{x}{2}\left(\sqrt{2 \cos ^2 x+3 \cos x}+\sqrt{\cos ^2 x+\sin x+4}\right)} \\& -\frac{1}{2 \sqrt{5}}\end{aligned}$