Question

If {x} denotes the fractional part of x, then $\underset{x\to 1}{lim}\frac{xsin\left\{x\right\}}{x-1}$, is

• A. 0
• B. -1
• C. non-existent
• D. none of these

Answer 1

Answer: (C)

non-existent

We have,
$\underset{x\to 1^{-}}{lim}\frac{xsin\left\{x\right\}}{x-1}=\underset{x\to 1^{-}}{lim}\frac{xsinx}{x-1}\to -\infty$ and,
$\underset{x\to 1^{+}}{lim}\frac{xsin\left\{x\right\}}{x-1}=\underset{x\to 1^{+}}{lim}\frac{xsin(x-1)}{x-1}=1\times 1=1$
Clearly, $\underset{x\to 1^{-}}{lim}\frac{xsin\left\{x\right\}}{x-1}\ne \underset{x\to 1^{+}}{lim}\frac{xsin\left\{x\right\}}{x-1}$
So, $\underset{x\to 1}{lim}\frac{xsin\left\{x\right\}}{x-1}$ does not exist.