Question

write the equation of all horizontal asymptotes of the function $f(x)=\frac{6x^{3}+3x^{5}}{6 - 5x^{5}}$.

Explanation:

Step1: Determine the degrees of numerator and denominator

The degree of the numerator $6x^{3}+3x^{2}$ is $n = 3$ (highest - power of $x$), and the degree of the denominator $6 - 5x^{5}$ is $m=5$.

Step2: Apply the horizontal - asymptote rule

When $n\lt m$, the horizontal asymptote of the rational function $y = \frac{f(x)}{g(x)}$ is $y = 0$. Since $3\lt5$ for the function $f(x)=\frac{6x^{3}+3x^{2}}{6 - 5x^{5}}$, the horizontal asymptote is $y = 0$.

Answer:

$y = 0$