Question

$\underset{x\to 1}{lim}\frac{x\sin (x-[x\left]\right)}{x-1}$, where [.] denotes the greatest integer function, is equal to

• A. $0$
• B. $-1$
• C. Non-existent
• D. None of these

Answer 1

The correct option is C Non-existent

$\underset{x\to 1}{lim}\frac{x\sin (x-[x\left]\right)}{x-1}$

put $x=1-h$ when$x\to 0,h\to 0$

$\text{L.H.L}=\underset{h\to 0}{lim}\frac{(1-h)\sin (1-h-[1-h])}{(1-h)-1}=\underset{h\to 0}{lim}\frac{(1-h)\sin (1-h)}{-h}=-\infty \text{R.H.L}=\underset{h\to 0}{lim}\frac{(1+h)\sin (1+h-[1+h])}{(1+h)-1}=\underset{h\to 0}{lim}\frac{(1+h)\sin h}{h}=1\text{Hence, the limit does not exist.}$

$sinceR.H.L\ne L.H.L$