Question

Is limit of the function $f\left(x\right)=\frac{x+|x|}{x}$ at $x=0$?

Answer 1

$f\left(x\right)=\frac{x+|x|}{x}$
Right limit at $x=0$
$R.H.L.=f(0+0)$
$=\underset{h\to 0}{lim}f(0+h)$
$=\underset{h\to 0}{lim}\frac{0+h+|0+h|}{0+h}(h>0)$
$=\underset{h\to 0}{lim}\frac{h+|h|}{h}$
$=\underset{h\to 0}{lim}\frac{h+h}{h}$
$=\underset{h\to 0}{lim}\frac{2h}{h}=2....\left(i\right)$
Left limit at $x=0$
$L.H.L.=f(0-0)$
$=\underset{h\to 0}{lim}f(0-h)$
$=\underset{h\to 0}{lim}\frac{0-h-(0-h)}{(0-h)}(h<0)$
$=\underset{h\to 0}{lim}\frac{-h+h}{-h}$
$=\underset{h\to 0}{lim}\frac{0}{-h}=\infty ...\left(ii\right)$
It is clear from equation (i) and (ii),
$L.H.L.\ne R.H.L.$
$\Rightarrow f(0-0)\ne x(0+0)$
Hence, limit of function does not exist at $x=0$.