x 0lim x(√2 ^2x+3 x- ^2x+ x+4) is:

Question

$\underset{x	o 0}{lim}\csc x(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4})$ is:

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}\csc x(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}) \ \underset{x	o 0}{lim}\frac{\csc ({\cos }^{2}x+3\cos x-\sin x-4)}{(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4})} \ \underset{x	o 0}{lim}\frac{1}{\sin x}\frac{({\cos }^{2}x+3\cos x-4)-\sin x}{(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4})} \ \underset{x	o 0}{lim}\frac{({\cos }^{x}+4)(\cos x-1)-\sin x}{\sin x(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4})} \ \underset{x	o 0}{lim}\frac{-2{\sin }^{2}\frac{x}{2}(\cos x+4)-2\sin \frac{x}{2}\cos \frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4})} \ \underset{x	o 0}{lim}\frac{-({\sin }^{x}\frac{x}{2}(\cos x+4)+\cos \frac{x}{2})}{\cos \frac{x}{2}(\sqrt{2{\cos }^{2}x+3\cos x}+\sqrt{{\cos }^{2}x+\sin x+4})} \ -\frac{1}{2\sqrt{5}} \end{gathered}$

Solutions · 1