_x 0 cosec x(√2 ^2 x+3 x- ^2 x+ x+4) is:

Question

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

Accepted Answer

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

Solutions · 1