part 1 of 5 recall that $|x + 4|=\begin{cases}x + 4&xgeq - 4\\-(x + 4)&x < - 4end{cases}$. therefore, we…

part 1 of 5 recall that $|x + 4|=\begin{cases}x + 4&xgeq - 4\\-(x + 4)&x < - 4end{cases}$. therefore, we will need to check the limits when approaching from the left and from the right. we will start by checking the limit when approaching from the left. as $x$ approaches $-4$ from the left, we have $lim_{x\rightarrow - 4^{-}}\frac{3x + 12}{|x + 4|}=lim_{x\rightarrow - 4^{-}}\frac{3x + 12}{square}$.

part 1 of 5 recall that $|x + 4|=\begin{cases}x + 4&xgeq - 4\\-(x + 4)&x < - 4end{cases}$. therefore, we will need to check the limits when approaching from the left and from the right. we will start by checking the limit when approaching from the left. as $x$ approaches $-4$ from the left, we have $lim_{x\rightarrow - 4^{-}}\frac{3x + 12}{|x + 4|}=lim_{x\rightarrow - 4^{-}}\frac{3x + 12}{square}$.

Answer

Explanation:

Step1: Factor the numerator

Factor (3x + 12) as (3(x + 4)). So we have (\lim_{x\rightarrow - 4^{-}}\frac{3x + 12}{|x + 4|}=\lim_{x\rightarrow - 4^{-}}\frac{3(x + 4)}{|x + 4|}).

Step2: Determine the value of (|x + 4|) for (x\rightarrow - 4^{-})

When (x\rightarrow - 4^{-}), (x<-4), so (|x + 4|=-(x + 4)). Then (\lim_{x\rightarrow - 4^{-}}\frac{3(x + 4)}{|x + 4|}=\lim_{x\rightarrow - 4^{-}}\frac{3(x + 4)}{-(x + 4)}).

Step3: Simplify the limit expression

Cancel out the ((x + 4)) terms (since (x\neq - 4) when taking the limit). We get (\lim_{x\rightarrow - 4^{-}}\frac{3(x + 4)}{-(x + 4)}=\lim_{x\rightarrow - 4^{-}}-3=-3).

Answer:

(-3)