Question

find the horizontal asymptote, if any, of the graph of the rational function.

f(x)=\frac{-5 x+7}{3 x+7}

select the correct choice below and, if necessary, fill in the answer box to complete your
choice.

a. the horizontal asymptote is
(type an equation. simplify your answer. use integers or fractions for any
numbers in the equation.)
b. there is no horizontal asymptote.

Explanation:

Step1: Determine the degrees of numerator and denominator

For the rational function \( f(x)=\frac{-5x + 7}{3x+7} \), the degree of the numerator \(n\) (highest power of \(x\) in the numerator) is \(n = 1\) (since the highest - power term is \(-5x\)), and the degree of the denominator \(m\) (highest power of \(x\) in the denominator) is \(m = 1\) (since the highest - power term is \(3x\)).

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) of a rational function \(y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), the horizontal asymptote is given by \(y=\frac{a_n}{b_m}\).
Here, \(a_n=-5\) (the leading coefficient of the numerator) and \(b_m = 3\) (the leading coefficient of the denominator). So, \(y=\frac{-5}{3}\).

Answer:

A. The horizontal asymptote is \(y =-\frac{5}{3}\)