I believe you're asking about the one-sided limit,
[tex]\displaystyle \lim_{x\to-4^-}\frac{|x+4|}{x+4}[/tex]
Recall the definition of absolute value:
• [tex]|x| = x[/tex] if [tex]x\ge0[/tex]
• [tex]|x| = -x[/tex] if [tex]x < 0[/tex]
Since we're approaching -4 from the left, we're effectively focusing on a domain of [tex]x<-4[/tex] or [tex]x+4<0[/tex]. So, by the definition above, we have [tex]|x+4| = -(x+4)[/tex]. Then in the limit, we have
[tex]\displaystyle \lim_{x\to-4^-}\frac{|x+4|}{x+4} = \lim_{x\to-4^-}\frac{-(x+4)}{x+4} = \lim_{x\to-4^-}(-1) = \boxed{-1}[/tex]
