Review - Fall \( 2024-5 \% \) Find the horizontal asymptote, if any, of the graph of the rational function. \( h(x)=\frac{20 x^{3}}{5 x^{2}+1} \)

To find the horizontal asymptote of the rational function \( h(x) = \frac{20x^3}{5x^2 + 1} \), we look at the degrees of the polynomial in the numerator and the polynomial in the denominator. The numerator has a degree of 3, while the denominator has a degree of 2. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote for this function, indicating that as \( x \) approaches infinity or negative infinity, the function will also tend to infinity. However, if we consider vertical asymptotes, we would set the denominator equal to zero (\( 5x^2 + 1 = 0 \)), but in this case, there are no real solutions, as \( 5x^2 + 1 \) is always positive for all real \( x \).