6) $lim_{x\rightarrow - 5}\frac{x + 5}{|x + 5|}$

Explanation:

To evaluate the limit $\lim_{x\to-5} \frac{x+5}{|x+5|}$, we need to consider the left-hand limit and the right-hand limit because of the absolute value function.

Step1: Evaluate the right-hand limit

As $x \to -5^+$, $x$ is slightly greater than $-5$. This means $x+5 > 0$. Therefore, $|x+5| = x+5$. $$ \lim_{x\to-5^+} \frac{x+5}{|x+5|} = \lim_{x\to-5^+} \frac{x+5}{x+5} = \lim_{x\to-5^+} 1 = 1 $$

Step2: Evaluate the left-hand limit

As $x \to -5^-$, $x$ is slightly less than $-5$. This means $x+5 < 0$. Therefore, $|x+5| = -(x+5)$. $$ \lim_{x\to-5^-} \frac{x+5}{|x+5|} = \lim_{x\to-5^-} \frac{x+5}{-(x+5)} = \lim_{x\to-5^-} -1 = -1 $$

Step3: Compare the one-sided limits

Since the right-hand limit ($1$) is not equal to the left-hand limit ($-1$), the overall limit does not exist. $$ \lim_{x\to-5^+} \frac{x+5}{|x+5|} \neq \lim_{x\to-5^-} \frac{x+5}{|x+5|} $$

Answer:

The limit does not exist.