Problem
Find the following limit or state that it does not exist. \[ \lim _{x \rightarrow 12} \frac{x^{2}-144}{12-x} \]
Solution
Step 1 ：The function is not defined at x=12, but we can simplify the expression and then evaluate the limit.
Step 2 ：The numerator can be factored as a difference of squares: \(x^{2}-144 = (x-12)(x+12)\).
Step 3 ：We can cancel out the common factor in the numerator and denominator, which simplifies the function to \(-x - 12\).
Step 4 ：Substitute x=12 into the simplified function, we get \(-12 - 12 = -24\).
Step 5 ：Thus, the limit of the function as x approaches 12 is -24.
Step 6 ：Final Answer: The limit is \(\boxed{-24}\).
