Suppose that \[ -x+1 \leq g(x) \leq \frac{5 x+31}{x+6} \] for all $x$ in the interval $(-6,-4)$. Find $\lim _{x \rightarrow-5} g(x)$ If the limit does not exist, click on "Does Not Exist."

Step 1 ：Given that the function \(g(x)\) is bounded by two functions, \(-x+1\) and \(\frac{5x+31}{x+6}\), for all \(x\) in the interval \((-6,-4)\).

Step 2 ：We are asked to find the limit of \(g(x)\) as \(x\) approaches -5.

Step 3 ：We can find the limits of these two functions as \(x\) approaches -5 and see if they are equal.

Step 4 ：If they are, then that is the limit of \(g(x)\). If they are not, then the limit of \(g(x)\) does not exist.

Step 5 ：Let's calculate the limits of the bounding functions as \(x\) approaches -5.

Step 6 ：For the function \(-x+1\), the limit as \(x\) approaches -5 is \(1 - (-5) = 6\).

Step 7 ：For the function \(\frac{5x+31}{x+6}\), the limit as \(x\) approaches -5 is also \(6\).

Step 8 ：Since the limits of both bounding functions as \(x\) approaches -5 are equal, the limit of \(g(x)\) as \(x\) approaches -5 exists and is equal to this common value.

Step 9 ：Final Answer: The limit of \(g(x)\) as \(x\) approaches -5 is \(\boxed{6}\).