Question

Evaluate $\underset{x\to \frac{5\pi }{4}}{lim}[sinx+cosx],$ where [.] denotes the greatest integer function.

• A. $-2$
• B. $-1$
• C. $0$
• D. $1$

Answer 1

The correct option is A $-2$

$\sin x+\cos x=\sqrt{2}(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x)$

$=\sqrt{2}(\cos \frac{\pi }{4}\sin x+\sin \frac{\pi }{4}\cos x)$

$=\sqrt{2}\sin (x+\frac{\pi }{4})$

${lim}_{x\to \frac{5\pi }{4}}(\sin x+\cos x)$

$=\sqrt{2}{lim}_{x\to \frac{5\pi }{4}}\left(\sin (x+\frac{\pi }{4})\right)$

$=\sqrt{2}\left(\sin (\frac{5\pi }{4}+\frac{\pi }{4})\right)$

$=\sqrt{2}\left(\sin \left(\frac{6\pi }{4}\right)\right)$

$=\sqrt{2}\left(\sin \left(\frac{3\pi }{2}\right)\right)$

$=\sqrt{2}\left(\sin (\pi +\frac{\pi }{2})\right)$

$=-\sqrt{2}\left(\sin \frac{\pi }{2}\right)$

$=-\sqrt{2}$

${lim}_{x\to \frac{5\pi }{4}}[\sin x+\cos x]$

$=[-\sqrt{2}]$

$=[-1.414]=-2$