Question

what is a horizontal asymptote of the function below?
$f(x)=\frac{7 x^{3}-3 x + 7}{5 x^{2}-6 x - 6}$
a. $x = 0$
b. $y = 0$
c. there is no horizontal asymptote.
d. $y = \frac{7}{5}$

Explanation:

Step1: Recall the rule for horizontal asymptotes of rational functions

For a rational function \( f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n>m\), there is no horizontal asymptote. Here, \(n = 3\) (degree of the numerator \(7x^3-3x + 7\)) and \(m=2\) (degree of the denominator \(5x^2-6x - 6\)).

Answer:

C. There is no horizontal asymptote.