Question

determine if the following limit exists. compute the limit if it exists.
lim(x→ - 8) (x² - 64)/(x + 8)
select the correct choice below and fill in any answer boxes in your choice.
a. lim(x→ - 8) (x² - 64)/(x + 8) =
b. the limit does not exist.

Explanation:

Step1: Factor the numerator

We know that $x^{2}-64=(x + 8)(x - 8)$ by the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 8$. So the limit becomes $\lim_{x
ightarrow - 8}\frac{(x + 8)(x - 8)}{x + 8}$.

Step2: Simplify the function

Cancel out the common factor $(x + 8)$ in the numerator and denominator. We get $\lim_{x
ightarrow - 8}(x - 8)$.

Step3: Evaluate the limit

Substitute $x=-8$ into $x - 8$. So $(-8)-8=-16$.

Answer:

A. $\lim_{x
ightarrow - 8}\frac{x^{2}-64}{x + 8}=-16$