Question

find the equation of all horizontal asymptotes (if any) of the rational function.

f(x)=\frac{4 x^{2}-3 x - 2}{17 x^{2}+x - 1}

select the correct choice below and fill in any answer boxes within your choice

a. the function has one horizontal asymptote,
(simplify your answer type an equation. use integers or fractions for any numbers in the equation)
b. 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 equations)
c. the function has no horizontal asymptotes.

Explanation:

Step1: Determine the degrees of numerator and denominator

For the rational function \(f(x)=\frac{4x^{2}-3x - 2}{17x^{2}+x - 1}\), the degree of the numerator \(n = 2\) (since the highest - power term is \(4x^{2}\)) and the degree of the denominator \(m=2\) (since the highest - power term is \(17x^{2}\)).

Step2: Use the horizontal asymptote rule for \(n = m\)

When \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the horizontal asymptote is given by \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator.

Here, \(a_{n}=4\) (the coefficient of \(x^{2}\) in the numerator) and \(b_{m}=17\) (the coefficient of \(x^{2}\) in the denominator). So, \(y = \frac{4}{17}\).

Answer:

A. The function has one horizontal asymptote, \(y=\frac{4}{17}\)