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^-)(|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<-4` or
`x+4<0`. So, by the definition above, we have
`|x+4| = -(x+4)`. Then in the limit, we have

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