Question

Limitation of -4 to the left of the absolute value of x+4 divided by x+4

Answer 1

I believe you're asking about the one-sided limit,

$\displaystyle \lim_{x\to-4^-}\frac{|x+4|}{x+4}$

Recall the definition of absolute value:

• $|x| = x$ if $x\ge0$

• $|x| = -x$ if $x < 0$

Since we're approaching -4 from the left, we're effectively focusing on a domain of $x$ or $x+4$. So, by the definition above, we have $|x+4| = -(x+4)$. Then in the limit, we have

$\displaystyle \lim_{x\to-4^-}\frac{|x+4|}{x+4} = \lim_{x\to-4^-}\frac{-(x+4)}{x+4} = \lim_{x\to-4^-}(-1) = \boxed{-1}$