Question

If $f\left(x\right)=\{\begin{array}{c}\frac{\sin \left[x\right]}{\left[x\right]},\left[x\right]\ne 0\\ 0,\left[x\right]=0\end{array}$, where [.] denotes the greatest integer function, then ${lim}_{x\to 0}f\left(x\right)$ is equal to

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

Answer 1

The correct option is D none of these

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}\frac{\sin \left[x\right]}{\left[x\right]}$

$=\underset{x\to 0^{-}}{lim}\frac{\sin -1}{-1}=1$

And

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{\sin 0}{\left[x\right]}=0$