Question

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.
$lim_{x
ightarrow8}\frac{x^{2}+3x - 88}{x^{2}-64}$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $lim_{x
ightarrow8}\frac{x^{2}+3x - 88}{x^{2}-64}=square$ (type an integer or a simplified fraction.)
b. the limit does not exist

Explanation:

Step1: Factor the numerator and denominator

Factor $x^{2}+3x - 88=(x + 11)(x - 8)$ and $x^{2}-64=(x + 8)(x - 8)$. So the function becomes $\lim_{x
ightarrow8}\frac{(x + 11)(x - 8)}{(x + 8)(x - 8)}$.

Step2: Cancel out the common factor

Cancel out the common factor $(x - 8)$ (since $x
eq8$ when taking the limit), we get $\lim_{x
ightarrow8}\frac{x + 11}{x + 8}$.

Step3: Substitute $x = 8$

Substitute $x = 8$ into $\frac{x + 11}{x + 8}$, we have $\frac{8+11}{8 + 8}=\frac{19}{16}$.

Answer:

A. $\lim_{x
ightarrow8}\frac{x^{2}+3x - 88}{x^{2}-64}=\frac{19}{16}$