Suppose that
\[
\frac{-2 x-11}{x+5} \leq g(x) \leq x^{2}+9 x+17
\]
for all $x$ in the interval $(-5,-3)$.
Find $\lim _{x \rightarrow-4} g(x)$.
If the limit does not exist, click on "Does Not Exist."

Step 1 ：We are given that the function g(x) is bounded by two functions, \(\frac{-2 x-11}{x+5}\) and \(x^{2}+9 x+17\), on the interval (-5,-3).

Step 2 ：We need to find the limit of g(x) as x approaches -4. Since -4 is within the interval (-5,-3), we can find the limits of the bounding functions at x=-4 and see if they are equal.

Step 3 ：If the limits of the bounding functions are equal, then that is the limit of g(x). If they are not, then the limit of g(x) does not exist.

Step 4 ：Let's find the limits of the bounding functions at x=-4. For the function \(\frac{-2 x-11}{x+5}\), the limit as x approaches -4 is -3.

Step 5 ：For the function \(x^{2}+9 x+17\), the limit as x approaches -4 is also -3.

Step 6 ：The limits of both bounding functions at x=-4 are equal, which means the limit of g(x) as x approaches -4 exists and is equal to the common limit of the bounding functions.

Step 7 ：Final Answer: The limit of g(x) as x approaches -4 is \(\boxed{-3}\).