8) find the horizontal asymptote(s). f(x) = (x + 5)/(x^3 + 27) y = -3 y = -5 y = 0 y = 3 9) choose the best…

Explanation:

Step1: Recall horizontal - asymptote rules

For a rational function $f(x)=\frac{N(x)}{D(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n\lt m$, the horizontal asymptote is $y = 0$; if $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$; if $n>m$, there is no horizontal asymptote.

Step2: Analyze the first function $f(x)=\frac{x + 5}{x^3+27}$

Here, $N(x)=x + 5$ (degree $n = 1$) and $D(x)=x^3+27$ (degree $m = 3$). Since $n=1\lt m = 3$, the horizontal asymptote is $y = 0$.

Step3: Analyze the second function $f(x)=\frac{3x^3+5x^2-8x + 2}{4x^3-2x^2+6x - 1}$

Here, $N(x)=3x^3+5x^2-8x + 2$ (degree $n = 3$) and $D(x)=4x^3-2x^2+6x - 1$ (degree $m = 3$). Since $n=m = 3$, and $a_n = 3$, $b_m=4$, the horizontal asymptote is $y=\frac{3}{4}$. But we are only asked for the first - function's answer in the multiple - choice part.

Answer:

C. $y = 0$