Question

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

f(x)=\frac{8 x^{2}-5 x-6}{17 x^{2}+7 x-4}

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 equations )
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: 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 \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).

The degree of the numerator \(n = 2\) (since the leading term of \(8x^{2}-5x - 6\) is \(8x^{2}\)) and the degree of the denominator \(m=2\) (since the leading term of \(17x^{2}+7x - 4\) is \(17x^{2}\)).

Step2: Calculate the value of the horizontal asymptote

We use the formula \(y=\frac{a_n}{b_m}\), where \(a_n = 8\) (the coefficient of \(x^{2}\) in the numerator) and \(b_m=17\) (the coefficient of \(x^{2}\) in the denominator). So \(y=\frac{8}{17}\).

Answer:

B. The function has one horizontal asymptote, \(y = \frac{8}{17}\)