Question

Write the value of ${lim}_{x\to 2}\frac{|x-2|}{x-2}.$

Answer 1

${lim}_{x\to 2}\frac{|x-2|}{x-2}$
$LHL={lim}_{x\to 2^{-}}\frac{-(x-2)}{(x-2)}$
$={lim}_{h\to 0}\frac{2-(2-h)}{(2-h)-2}$
$={lim}_{h\to 0}\frac{h}{-h}$
$=-1$
$RHL={lim}_{x\to 2^{+}}\frac{(x-2)}{(x-2)}$
$=1$
$LHL=RHL$
$\therefore {lim}_{x\to 2}\frac{|x-2|}{(x-2)}.$ does not exist.