Question

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

• A. - 1
• B. Not existent
• C. None of these

Answer 1

The correct option is B Not existent

$\underset{x\to 1}{lim}\frac{xsin\{x-[x\left]\right\}}{x-1}RHL=\underset{h\to 0}{lim}\frac{(1+h)sin\{1+h-[1+h\left]\right\}}{1+h-1}=\underset{h\to 0}{lim}\frac{(1+h)sin(1+h-1)}{h}=1LHL=\underset{h\to 0}{lim}\frac{(1-h)sin\{1-h-[1-h\left]\right\}}{1-h-1}=\underset{h\to 0}{lim}\frac{(1-h)sin(1-h)}{-h}=\infty LHL\ne RHL$