Let $f(x)=\frac{x^{2}-5 x-24}{x-8}$. Find a) $\lim _{x \rightarrow 8} f(x)$

Step 1 ：Let \(f(x)=\frac{x^{2}-5 x-24}{x-8}\). We need to find \(\lim _{x \rightarrow 8} f(x)\).

Step 2 ：The function is undefined at x=8, so we can't just plug in the value.

Step 3 ：However, we can simplify the function by factoring the numerator and cancelling out the common factor with the denominator. This will give us a new function that is equivalent to the original for all x except x=8.

Step 4 ：We then find the limit as x approaches 8 of this new function.

Step 5 ：The simplified function is \(x + 3\).

Step 6 ：The limit of the function as x approaches 8 is 11. This makes sense because the function is continuous at all points except x=8, and the limit as x approaches 8 from both the left and the right is the same.

Step 7 ：Final Answer: The limit of the function as x approaches 8 is \(\boxed{11}\).