Questions: Find the given limit. lim as x approaches -9 of (x^2-5)/(9-x) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The limit as x approaches -9 of (x^2-5)/(9-x) = (Simplify your answer.) B. The limit does not exist.

Find the given limit.
lim as x approaches -9 of (x^2-5)/(9-x)

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The limit as x approaches -9 of (x^2-5)/(9-x) = (Simplify your answer.)
B. The limit does not exist.

Transcript text: Find the given limit. \[ \lim _{x \rightarrow-9} \frac{x^{2}-5}{9-x} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow-9} \frac{x^{2}-5}{9-x}=$ $\square$ (Simplify your answer.) B. The limit does not exist.

Solution

Solution Steps

To find the limit of the given function as $ x $ approaches -9, we can directly substitute $ x = -9 $ into the function if it does not result in an indeterminate form. If it does, we need to simplify the expression first.

Step 1: Define the Function

We start with the function: \[ f(x) = \frac{x^2 - 5}{9 - x} \]

Step 2: Substitute $ x = -9 $

We need to find the limit as $ x $ approaches $-9$: \[ \lim_{x \to -9} \frac{x^2 - 5}{9 - x} \]

Step 3: Simplify the Expression

Substituting $ x = -9 $ directly into the function: \[ \frac{(-9)^2 - 5}{9 - (-9)} = \frac{81 - 5}{9 + 9} = \frac{76}{18} = \frac{38}{9} \]