Question

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

Explanation:

Step1: Determine the degrees of numerator and denominator

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

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

When \(n = m\), the horizontal asymptote \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator.
For the numerator \(5x - 2x^{3}+4\), the leading coefficient \(a_{n}=-2\).
For the denominator \(3x^{3}-5x^{2}+1\), the leading coefficient \(b_{m}=3\).
So \(y = \frac{-2}{3}\).

Answer:

\(y=-\frac{2}{3}\)