find the horizontal asymptote of f(x) = (3x - 4x^3 + x^2 + 10)/(2 + x + x^2 + 8x^3).

Explanation:

Step1: Identify degrees of polynomials

The degree of the numerator $n = 3$ (highest - power of $x$ is $x^{3}$ in $-4x^{3}$) and the degree of the denominator $m = 3$ (highest - power of $x$ is $x^{3}$ in $8x^{3}$).

Step2: Use the horizontal - asymptote rule for equal degrees

When $n = m$, the horizontal asymptote $y$ is the ratio of the leading coefficients. The leading coefficient of the numerator $a=-4$ and the leading coefficient of the denominator $b = 8$. $y=\frac{a}{b}$ $y=\frac{-4}{8}=-\frac{1}{2}$

Answer:

$y =-\frac{1}{2}$