Problem
The initial substitution of $x=a$ yields the form $\frac{0}{0}$. Simplify the function algebraically, or use a table or graph to determine the limit. If necessary, state that the limit does not exist. \[ \lim _{x \rightarrow 3} \frac{2 x^{2}+6 x-36}{x^{2}-9} \] Select the correct choice below and, if necessary, fill in the answer box to complete youtr choice. A. $\lim _{x \rightarrow 3} \frac{2 x^{2}+6 x-36}{x^{2}-9}=\square$ (Type an integer or a simplified fraction.) B. The limit does not exist.
Solution
Step 1 ：The function is in the form of \(\frac{0}{0}\) when \(x=3\). This is an indeterminate form, so we can't directly substitute \(x=3\) into the function to find the limit.
Step 2 ：However, we can simplify the function by factoring the numerator and the denominator. The function \(\frac{2 x^{2}+6 x-36}{x^{2}-9}\) simplifies to \(\frac{2(x + 6)}{x + 3}\).
Step 3 ：After simplifying, we can substitute \(x=3\) into the simplified function to find the limit.
Step 4 ：The limit of the function as \(x\) approaches 3 is \(\boxed{3}\).
