Question

find all horizontal asymptotes of the following function.

$f(x)=\frac{2(x + 2)(x - 2)}{3(2x - 3)(x + 3)}$

Explanation:

Step1: Expand numerator and denominator

Expand \(2(x + 2)(x - 2)=2(x^{2}-4)=2x^{2}-8\)
Expand \(3(2x - 3)(x + 3)=3(2x^{2}+6x-3x - 9)=3(2x^{2}+3x - 9)=6x^{2}+9x - 27\)
So \(f(x)=\frac{2x^{2}-8}{6x^{2}+9x - 27}\)

Step2: Use the rule for horizontal asymptotes of rational functions

For a rational function \(y = \frac{f(x)}{g(x)}=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots+b_{0}}\), if \(n = m\), the horizontal asymptote is \(y=\frac{a_{n}}{b_{m}}\)
Here \(n = 2\), \(m = 2\), \(a_{2}=2\), \(b_{2}=6\)

Answer:

The horizontal asymptote is \(y=\frac{1}{3}\)