Question

find the horizontal asymptote of ( f(x)=\frac{-2 x-5 x^{3}+4}{3 x^{3}-3 x^{2}-1} ).
( y= )

Explanation:

Step1: Determine the degrees of numerator and denominator

The degree of the numerator \( -2x - 5x^{3}+4\) is \(n = 3\) (since the highest - power term is \(-5x^{3}\)), and the degree of the denominator \(3x^{3}-3x^{2}-1\) is \(m = 3\).

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

When \(n=m\), the horizontal asymptote \(y\) is given by the ratio of the leading coefficients. The leading coefficient of the numerator \(a=-5\) (from \(-5x^{3}\)) and the leading coefficient of the denominator \(b = 3\) (from \(3x^{3}\)). So, \(y=\frac{a}{b}\).

Answer:

\(y =-\frac{5}{3}\)