lim x→5− |x−5| / x−5

Explanation:

Step1: Analyze the absolute - value for $x\to5 -$

When $x\to5 -$, that is $x\lt5$, then $|x - 5|=-(x - 5)$.

Step2: Substitute the absolute - value expression

Substitute $|x - 5|=-(x - 5)$ into the limit $\lim_{x\to5 -}\frac{|x - 5|}{x - 5}$, we get $\lim_{x\to5 -}\frac{-(x - 5)}{x - 5}$.

Step3: Simplify the expression

$\frac{-(x - 5)}{x - 5}=-1$ for $x\neq5$. So $\lim_{x\to5 -}\frac{|x - 5|}{x - 5}=-1$.

Answer:

$-1$