Question

${lim}_{x\to k}(x-[x\left]\right)$, where $k$ is an integer, is equal to (where $[\cdot ]$ denotes greatest integer function).

• A. $1$
• B. $0$
• C. $-1$
• D. $Doesnotexist$

Answer 1

The correct option is D $Doesnotexist$

$\sin \alpha +\sin (\alpha +\frac{2\pi }{3})+\sin (\alpha +\frac{4\pi }{3})=L$

$L=\sin \alpha +2\sin \left(\frac{2\alpha +7\pi }{2}\right)\cos \left(\frac{\frac{4\pi }{3}+\alpha -\alpha -\frac{2\pi }{3}}{2}\right)$

$L=\sin \alpha +2\sin (\alpha +\pi )\cos \frac{\pi }{3}$

$L=\sin \alpha +2(-\sin \alpha )\times \frac{1}{2}$

$L=0$

$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2}$

$\sin (\alpha +\pi )=-\sin a$

Hence:-

$\sin \alpha +\sin (\alpha +\frac{2\pi }{3})+\sin (\alpha +\frac{4\pi }{3})=0$