Question

A function $f\left(x\right)=\left\{\frac{\sin 3x}{x3}\right\}\frac{x\ne 0}{x=0}$ , find whether function $f\left(x\right)$ is continuous at $x=0$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\sin 3x,x\ne 0\\ x^3,x=0\end{array}$

Checking continuity at $x=0$

$\Rightarrow {lim}_{x\to 0^{+}}\sin 3\left(x\right)={lim}_{h\to 0}\sin 3(0+h)=0\to 1$

$\Rightarrow {lim}_{x\to 0^{-}}\sin 3\left(x\right)={lim}_{h\to 0}\frac{\sin \left(3\right)(0-h)}{1}=0\to 2$

$\Rightarrow f\left(0\right)={\left(0\right)}^3\to 3$

From $1,2\&3$

${lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{+}}f\left(x\right)=f\left(0\right)=0$

$\Rightarrow f\left(x\right)$ is continuous at $x=0$