Question

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

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

Answer 1

The correct option is D does not exist

The given function can be restated as
$f\left(x\right)=\{\begin{array}{ccc}\frac{\sin \left[x\right]}{\left[x\right]}& if& xϵ(-\infty ,0)\cup [1,\infty ]\\ 0& if& xϵ[0,1)\end{array}$
$\therefore \underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}\frac{\sin [-h]}{[-h]}$
$=\underset{h\to 0}{lim}\frac{\sin (-1)}{(-1)}=\sin 1and\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}0=0$
$\because \underset{x\to 0^{+}}{lim}f\left(x\right)\ne \underset{x\to 0^{+}}{lim}f\left(x\right)$
$\therefore \underset{x\to 0}{lim}f\left(x\right)$ does not exist