Question

show that $\underset{x\to 4}{lim}\frac{\left|x-4\right|}{x-4}$ does not exists

Answer 1

$\underset{x\to 4}{lt}\frac{|x-4|}{x-4}$

$\underset{x\to 4^{+}}{lt}\frac{x-4}{x-4}=1$

$\underset{x\to 4^{-}}{lt}\frac{4-x}{x-4}=-1$

$\therefore \underset{x\to 4^{+}}{lt}\ne \underset{x\to 4^{-}}{lt}$

$\therefore$ Limit does not exists.