Question

Find $\underset{x\to 5}{\text{lim}}$, where f(x) = |x| - 5,

Answer 1

Here f(x) = |x| -5

L.H.L. = $\underset{x\to 5}{\text{lim}}f\left(x\right)=\underset{x\to 5^{-}}{\text{lim}}|x|-5$

Put x = 5 - h as $x\to 5,h\to 0$

$\therefore \underset{h\to 0}{\text{lim}}|5-h|-5=\underset{h\to 0}{\text{lim}}$ 5- h-5

= $\underset{h\to 0}{\text{lim}}$ (-h) = 0

R.H.L. = $\underset{x\to 5^{+}}{\text{lim}}f\left(x\right)=\underset{x\to 5^{+}}{\text{lim}}$ |x| -5

Put x = 5 + h as $x\to x\to 5,h\to 0$

$\therefore \underset{h\to 0}{|5+h|}-5=\underset{h\to 0}{\text{lim}}$ 5+h-5

= $\underset{h\to 0}{\text{lim}}$ h=0

Now L.H.L. = R.H.L.

Thus limit exists at x = 5 and $\underset{h\to 0}{\text{lim}}$ f(x) = 0.