Question

Find the indicated limit, if it exists. limit of f of x as x approaches 0 where f of x equals 5 x minus 8 when x is less than 0 and the absolute value of the quantity negative 4 minus x when x is greater than or equal to 0

Answer 1

$f(x)=\begin{cases}5x-8&\text{for }x$

The limits from either side are

$\displaystyle\lim_{x\to0^-}f(x)=\lim_{x\to0}(5x-8)=-8$

$\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0}|-4-x|=\lim_{x\to0}|x+4|=|4|=4$

The one-sided limits don't match, so the limit as $x\to0$ does not exist.