Answer:
[tex]\lim_{x \to 2^-} \frac{|x-2|}{x-2}=-1[/tex].
Step-by-step explanation:
We want to find [tex]\lim_{x \to 2^-} \frac{|x-2|}{x-2}[/tex].
By definition:
[tex]|x-2|=\left \{ {{x-2,\:if\:x\:>\:2} \atop {-(x-2),\:if\:x\:<\:2}} \right.[/tex]
Since we want to find the Left Hand Limit, we use f(x)=-(x-2)
[tex]\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=\lim_{x \to 2} \frac{-(x-2)}{x-2}[/tex].
[tex]\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=\lim_{x \to 2} (-1)[/tex].
The limit of a constant is the constant.
[tex]\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=-1[/tex].
