Question

Find $\underset{x\to 0}{\text{lim}}$ f(x) where f(x) = $\{\begin{array}{cc}\frac{x}{|x|}& x\ne 0\\ 0& x\ne 0\end{array}$

Answer 1

Here f(x) = $\{\begin{array}{cc}\frac{x}{|x|}& x\ne 0\\ 0& x\ne 0\end{array}$

L.H.L. = $\underset{x\to 0^{-}}{\text{lim}}f\left(x\right)=\underset{x\to 0^{-}}{\text{lim}}\frac{x}{|x|}$

Put x=0 - h as $x\to 0,h\to 0$

$\therefore \underset{h\to 0^{-}}{\text{lim}}\frac{-h}{|-h|}=\underset{h\to 0^{-}}{\text{lim}}\frac{-h}{|-h|}$

= $\underset{h\to 0^{-}}{\text{lim}}\frac{-h}{h}$ =- 1

R.H.L. = $\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)=\underset{x\to 0^{+}}{\text{lim}}\frac{x}{|x|}$

Put x=0 + h as $x\to x\to 0,h\to 0$

$\therefore \underset{h\to 0}{\text{lim}}\frac{0+h}{|0+h|}=\underset{h\to 0}{\text{lim}}\frac{h}{|h|}=\underset{h\to 0}{\text{lim}}\frac{h}{h}$=1

Now L.H.L. $\ne$ R.H.L.

Thus limit does not exist at x =0.