Question

The value of the limit
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$ is _______.

Answer 1

$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{4 \sqrt{2} \cdot 2 \sin 2 x \cos x}{2 \sin 2 x \sin \frac{3 x}{2}+\left(\cos \frac{5 x}{2}-\cos \frac{3 x}{2}\right)-\sqrt{2}(1+\cos 2 x)}\right) $
$\Rightarrow \displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{16 \sqrt{2} \sin x \cos ^{2} x}{8 \sin x \sin \frac{x}{2} \cdot \cos ^{2} x-2 \sqrt{2} \cos ^{2} x}\right)$
$\Rightarrow \displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{16 \sqrt{2} \sin x}{8 \sin x \sin \frac{x}{2}-2 \sqrt{2}}=8$