Answer
the limit does not exist
Work Step by Step
Theorem 1
$\displaystyle \lim_{x\rightarrow a}f(x)=L$ if and only if $\displaystyle \lim_{x\rightarrow a^{-}}f(x)=L=\lim_{x\rightarrow a^{+}}f(x)$
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$|x+6|=\left\{\begin{array}{lll}
x+6 & if & x \geq -6\\
-(x+6) & if & x < -6
\end{array}\right.$
Approaching x=-6 from the right,
$\displaystyle \lim_{x\rightarrow-6+}\frac{2x+12}{|x+6|}=\lim_{x\rightarrow-6+}\frac{2(x+6)}{x+6}=2$
Approaching x=-6 from the left,
$\displaystyle \lim_{x\rightarrow-6^{-}}\frac{2x+12}{|x+6|}=\lim_{x\rightarrow-6^{-}}\frac{2(x+6)}{-(x+6)}=-2$
The left and right limits exist, but are NOT EQUAL,
so by Th. 1,
$\displaystyle \lim_{x\rightarrow-6}\frac{2x+12}{|x+6|}$ does not exist.
