$\lim\limits_{x \rightarrow 5 \pi / 4}[\sin x+\cos x]$, where [.] denotes the integral part of x

Is equal to –2

Given limit is

$\lim\limits_{x \rightarrow \frac{5-\pi}{4}}\left[\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)\right]$

$\lim\limits_{x \rightarrow \frac{5-\pi}{4}+}\left[\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)\right]=-2, \lim\limits_{x \rightarrow \frac{5-\pi}{4}-}\left[\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)\right]=-2$

Hence (2) is the correct answer.