Question

what is a horizontal asymptote of the function below?
$f(x)=\frac{2 x^{2}+5 x + 9}{3 x^{3}+6 x - 9}$
a. there is no horizontal asymptote.
b. $y = 0$
c. $y = \frac{2}{3}$
d. $x = 0$

Explanation:

Step1: Recall the rule for horizontal asymptote of rational functions

For a rational function \(f(x)=\frac{a_nx^n +\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n\lt m\), then \(y = 0\) is the horizontal asymptote.

Step2: Identify the degrees of numerator and denominator

In the function \(f(x)=\frac{2x^{2}+5x + 9}{3x^{3}+6x-9}\), the degree of the numerator \(n = 2\) (since the highest - power term is \(2x^{2}\)) and the degree of the denominator \(m=3\) (since the highest - power term is \(3x^{3}\)).

Since \(n = 2\lt m = 3\)

Answer:

B. \(y = 0\)