Question

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin \left[x\right]}{\left[x\right]},& for,x\ne 0\\ 0,& for,x=0\end{array}$ is?

Answer 1

$f\left(x\right)=\{\begin{array}{c}\frac{\sin \left[x\right]}{\left[x\right]},x\ne 0\\ 0,x=0\end{array}$

Continuity $@0$

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

$\underset{x\to 0}{lim}=\frac{\sin [0-h]}{[0-h]}=\frac{\sin (-1)}{-1}=\sin 1-LHL$

$f\left(0\right)=0$ Since $LHS\ne RHS=f\left(0\right)$

then $f\left(x\right)$ not continuous $@x=0$

i.e Discontinuous at $x=0$

$LHL$ is $\sin 1$ (a well defined number)

$RHL$ does not exist, indeterminate