Question

question
find all horizontal asymptotes of the following function.
$f(x)=\frac{2(x + 7)(x - 7)}{3(5x - 8)(x - 8)}$
answer attempt 1 out of 2
one horizontal asymptote
no horizontal asymptotes
one horizontal asymptote
two horizontal asymptotes

Explanation:

Step1: Expand the numerator and denominator

First, expand $2(x + 7)(x - 7)=2(x^{2}-49)=2x^{2}-98$ and $3(5x - 8)(x - 8)=3(5x^{2}-40x-8x + 64)=15x^{2}-144x + 192$. So $f(x)=\frac{2x^{2}-98}{15x^{2}-144x + 192}$.

Step2: Use the rule for horizontal asymptotes of rational functions

For a rational function $y = \frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$, when $n = m$ (here $n = m=2$), the horizontal - asymptote is $y=\frac{a_n}{b_m}$. Here $a_n = 2$ and $b_m = 15$, so the horizontal asymptote is $y=\frac{2}{15}$.

Answer:

One Horizontal Asymptote with equation $y=\frac{2}{15}$