Question

find the horizontal and vertical asymptotes.
f(x)=\frac{9x^{2}+16}{x^{2}-16}
find the horizontal asymptote(s). select the correct choice below and fill in any answer boxes in your choice.
a. y=
(use a comma to separate answers as needed.)
b. there are no horizontal asymptotes.

Explanation:

Step1: Determine horizontal asymptote

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, when $n = m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$. Here, $f(x)=\frac{9x^2 + 16}{x^2-16}$, $n = m=2$, $a_n = 9$, $b_m=1$.

Step2: Calculate the value of horizontal asymptote

$y=\frac{9}{1}=9$.

Step3: Determine vertical asymptote

Set the denominator equal to zero. Solve $x^2 - 16=0$. Using the difference - of - squares formula $a^2 - b^2=(a + b)(a - b)$, we have $(x + 4)(x - 4)=0$.

Step4: Solve for x

$x=-4$ or $x = 4$.

Answer:

Horizontal asymptote: A. $y = 9$
Vertical asymptotes: $x=-4,x = 4$