Question

find the horizontal asymptote of ( f(x)=\frac{5 x+3 x^{3}+1}{-5 x^{3}-2 x^{2}-3} ). if the horizontal asymptote does not exist, enter dne. the horizontal asymptote is ( y= )

Explanation:

Step1: Identify the degrees of numerator and denominator

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

Step2: Use the horizontal - asymptote rule for rational functions

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 function \(f(x)=\frac{5x + 3x^{3}+1}{-5x^{3}-2x^{2}-3}\), \(a_{n}=3\) and \(b_{m}=-5\).
So, \(y = \frac{3}{-5}=-\frac{3}{5}\)

Answer:

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