Question

evaluate the limit, if it exists. if a limit does not exist, type dne. $lim_{x
ightarrow - 5}\frac{2x + 10}{|x + 5|}$ limit:

Explanation:

Step1: Factor the numerator

Factor out 2 from the numerator $2x + 10$, we get $2(x + 5)$. So the function becomes $\lim_{x
ightarrow - 5}\frac{2(x + 5)}{|x + 5|}$.

Step2: Consider left - hand and right - hand limits

Left - hand limit ($x

ightarrow - 5^{-}$):
When $x
ightarrow - 5^{-}$, $x+5<0$, so $|x + 5|=-(x + 5)$. Then $\lim_{x
ightarrow - 5^{-}}\frac{2(x + 5)}{|x + 5|}=\lim_{x
ightarrow - 5^{-}}\frac{2(x + 5)}{-(x + 5)}=-2$.

Right - hand limit ($x

ightarrow - 5^{+}$):
When $x
ightarrow - 5^{+}$, $x + 5>0$, so $|x + 5|=x + 5$. Then $\lim_{x
ightarrow - 5^{+}}\frac{2(x + 5)}{|x + 5|}=\lim_{x
ightarrow - 5^{+}}\frac{2(x + 5)}{x + 5}=2$.

Step3: Determine the limit

Since the left - hand limit $\lim_{x
ightarrow - 5^{-}}\frac{2(x + 5)}{|x + 5|}=-2$ and the right - hand limit $\lim_{x
ightarrow - 5^{+}}\frac{2(x + 5)}{|x + 5|}=2$ are not equal, the two - sided limit $\lim_{x
ightarrow - 5}\frac{2(x + 5)}{|x + 5|}$ does not exist.

Answer:

DNE