Questions: Use the Theorem on Limits of Rational Functions to find the limit. If necessary, state that the limit does not exist.
lim (x -> -8) (x^2 - 6) / (8 - x)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
lim (x -> -8) (x^2 - 6) / (8 - x) = (Simplify your answer.)
B. The limit does not exist.
Transcript text: Use the Theorem on Limits of Rational Functions to find the limit. If necessary, state that the limit does not exist.
\[
\lim _{x \rightarrow-8} \frac{x^{2}-6}{8-x}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
$\lim _{x \rightarrow-8} \frac{x^{2}-6}{8-x}=$ $\square$ (Simplify your answer.)
B. The limit does not exist.
Solution
Solution Steps
Step 1: Identify the function and the point of interest
The function given is \(f(x) = (x^2-6)/(8-x)\) and we are interested in the limit as \(x\) approaches -8.
Step 2: Check if direct substitution is possible
Direct substitution of \(x = -8\) into \(f(x)\) gives \(f(-8) = 29/8\), which is not indeterminate.
Step 3: Apply limit laws or simplify if necessary
No further simplification was necessary.
Final Answer:
The limit of \(f(x)\) as \(x\) approaches -8 is 3.625.