Use the Theorem on Limits of Rational Functions to find the limit. If necessary, state that the limit does not exist. \[ \lim _{x \rightarrow-8} \frac{x^{2}-1}{8-x} \]

Step 1 ：We are given the function \(f(x) = \frac{x^{2}-1}{8-x}\) and we are asked to find the limit as x approaches -8.

Step 2 ：The function is undefined at x = 8, but we are interested in the limit as x approaches -8, where the function is defined.

Step 3 ：So, we can directly substitute x = -8 into the function to find the limit.

Step 4 ：Substituting x = -8 into the function, we get \(f(-8) = \frac{(-8)^{2}-1}{8-(-8)} = \frac{63}{16}\).

Step 5 ：Hence, the limit of the function as x approaches -8 is \(\frac{63}{16}\).

Step 6 ：Final Answer: \(\boxed{\frac{63}{16}}\)