Question

find the equation of all horizontal asymptotes (if any) of the rational function
f(x)=\frac{6 x^{3}-x^{2}+1}{x^{2}+7}
select the correct choice below and fill in any answer boxes within your choice
a. the function has two horizontal asymptotes. the top asymptote is , and the bottom asymptote is
(simplify your answers. type equations. use integers or fractions for any numbers in the equation)
b. the function has one horizontal asymptote,
(simplify your answer. type an equation. use integers or fractions for any numbers in the equation)
c. the function has no horizontal asymptotes

Explanation:

Step1: Determine the degrees of numerator and denominator

The degree of the numerator \(n = 3\) (from \(6x^{3}-x^{2}+1\)), and the degree of the denominator \(m = 2\) (from \(x^{2}+7\)).

Step2: Apply the rule for horizontal asymptotes

When \(n>m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the rational function \(y = f(x)\) has no horizontal asymptotes.

Answer:

C. The function has no horizontal asymptotes.