Question

use properties of limits to find the indicated limit. it may be necessary to rewrite the expression before limit properties can be applied.
lim_{x \to 3} \frac{x^{2}+5x - 24}{x^{2}-9}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. lim_{x \to 3} \frac{x^{2}+5x - 24}{x^{2}-9}=\frac{11}{6} (type an integer or a simplified fraction.)
b. the limit does not exist and is neither (infty) nor (-infty).

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+5x - 24=(x + 8)(x-3)$ and the denominator $x^{2}-9=(x + 3)(x - 3)$. So the limit becomes $\lim_{x
ightarrow3}\frac{(x + 8)(x - 3)}{(x + 3)(x - 3)}$.

Step2: Cancel out the common factor

Since $x
eq3$ when taking the limit, we can cancel out the $(x - 3)$ terms. The limit simplifies to $\lim_{x
ightarrow3}\frac{x + 8}{x+3}$.

Step3: Substitute the value of x

Substitute $x = 3$ into $\frac{x + 8}{x+3}$, we get $\frac{3+8}{3 + 3}=\frac{11}{6}$.

Answer:

A. $\lim_{x
ightarrow3}\frac{x^{2}+5x - 24}{x^{2}-9}=\frac{11}{6}$