Question

Using the definition, show that the function is discontinuous at the point $x=0$.

$f\left(x\right)=\frac{\sin x}{|x|}$ if $x\ne 0,and$

$f\left(x\right)=1$ if $x=0$

Answer 1

$f\left(x\right)=\frac{\sin x}{|x|}x\ne 0$

Let check L.H.L. $\underset{x\to 0^{-}}{lim}\frac{\sin x}{|x|}$

$=\underset{x\to 0^{-}}{lim}\frac{\sin x}{-x}$ [for $X<0,|x|=-x]$

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

$=-1$

and R.H.L. $\underset{x\to 0^{+}}{lim}\frac{\sin x}{|x|}$

$\Rightarrow \underset{x\to 0^{+}}{lim}\frac{\sin x}{x}$ [for $x>0,|x|=x]$

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

$=1$

here $L.H.L.\ne R.H.L.$ so function is discontinuous at $x=0$