Evaluate the limit: \[ \lim _{x \rightarrow 9} \frac{-8 x+72}{x^{2}-12 x+27}= \]

Step 1 ：The limit of a function as x approaches a certain value can be found by substituting the value of x into the function. However, in this case, if we substitute x=9 into the function, we get a 0/0 form which is indeterminate. Therefore, we need to simplify the function first. We can do this by factoring the numerator and the denominator.

Step 2 ：Simplify the function to get \(-8/(x - 3)\).

Step 3 ：Calculate the limit as x approaches 9, we find that the limit is -4/3.

Step 4 ：Final Answer: The limit of the function as x approaches 9 is \(\boxed{-\frac{4}{3}}\).