Question

recall that
|x + 6|=\begin{cases}x + 6&xgeq - 6\\-(x + 6)&xlt - 6end{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 (-6) from the left, we have
(lim_{x
ightarrow - 6^{-}}\frac{5x + 30}{|x + 6|}=lim_{x
ightarrow - 6^{-}}\frac{5x + 30}{square}).

Explanation:

Step1: Simplify the numerator

Factor out 5 from $5x + 30$ to get $5(x + 6)$.

Step2: Determine the form of $|x + 6|$ for $x\to - 6^{-}$

Since $x\to - 6^{-}$ (approaching - 6 from the left), $x<-6$. So $|x + 6|=-(x + 6)$.

Step3: Substitute into the limit

$\lim_{x\to - 6^{-}}\frac{5x + 30}{|x + 6|}=\lim_{x\to - 6^{-}}\frac{5(x + 6)}{-(x + 6)}$.

Step4: Cancel out the common factor

Cancel out $(x + 6)$ (since $x
eq - 6$ when taking the limit), we get $\lim_{x\to - 6^{-}}\frac{5(x + 6)}{-(x + 6)}=\lim_{x\to - 6^{-}}-5=-5$.

Answer:

$-5$