Final answer:
The limit as x approaches -9 of the given expression is -1/81 after simplifying and cancelling terms; there is no division by zero, so the limit exists.
Step-by-step explanation:
We are tasked to evaluate the limit as x approaches -9 of the following expression:
\(\lim_{x\to -9}\frac{\frac{1}{9} + \frac{1}{x}}{9+x}\)
Let's start by simplifying the expression inside the limit:
\(\frac{1}{9} + \frac{1}{x} = \frac{x+9}{9x}\)
The limit now becomes:
\(\lim_{x\to -9}\frac{x+9}{9x(9+x)}\)
Cancelling out the (x+9) terms, we get:
\(\lim_{x\to -9}\frac{1}{9x}\)
The direct answer in 2 lines would be:
Plugging in x = -9:
\(\frac{1}{9(-9)} = -\frac{1}{81}\)
Since the expression simplifies to a fraction with no division by zero, the limit exists and is equal to -1/81.