Question

If
$f\left(x\right)=\{\begin{array}{cc}\frac{{\left[x\right]}^2+\sin \left[x\right]}{\left[x\right]}& \text{for}\left[x\right]\ne 0\\ 0& \text{for}\left[x\right]=0\end{array}$
where $\left[x\right]$ denotes the greatest integer less than or equal to x, then ${lim}_{x\to 0}f\left(x\right)$ equals

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

Answer 1

The correct option is D limit does not exist.

As $x\to 0$- (i.e., approaches 0 from the left), $\left[x\right]=-1$

$\therefore \underset{x\to 0^{-}}{lim}\frac{1+\sin (-1)}{-1}=-1+\sin 1$

whereas, if $x\to 0^{+}$, we get $\left[x\right]=0$

$\therefore f\left(x\right)=0\Rightarrow \underset{x\to 0^{+}}{lim}f\left(x\right)=0$

Thus $\underset{x\to 0}{lim}f\left(x\right)$ does not exist.