Question

Consider $f\left(x\right)=sgn\left(\sin x\right)+\left[x\right];2\le x\le 4$ and $g\left(x\right)=-2+|x-3|;$ where [ ] denotes greatest integer function. Then $\underset{x\to 3}{lim}gof\left(x\right)$ is equal to

• A. $-1$
• B. $0$
• C. $1$
• D. Does not exist

Answer 1

Answer: (D)

Does not exist

$f\left(x\right)=sgn\left(\sin x\right)+\left[x\right]:2\le x\le 4$

$g\left(x\right)=-2+[x-3]$

$g\left(f\left(x\right)\right)=-2+[sgn\left(\sin x\right)+\left[x\right]-3]$

$\underset{x\to 3^{+}}{lim}gof\left(x\right)=-2+[+1+3-3][\sin 3>0];\left[3.0001\right]=3$

$=-1$

$\underset{x\to 3^{-}}{lim}gof\left(x\right)=-2+[+1+2-3](\left[2.9999\right]=2)$

$=-2$

LHL$\ne$RHL

$\therefore \underset{x\to 3}{lim}gof\left(x\right)$ does not exist.