Question

determine the horizontal asymptote of the function. if none exists, state that fact.

f(x)=\frac{3 x}{x^{3}-8 x^{2}}

select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

a. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations.)

b. the function has one horizontal asymptote, (type an equation.)

c. the function has no horizontal asymptotes.

Explanation:

Step1: Simplify the function

First, simplify \(f(x)=\frac{3x}{x^{3}-8x^{2}}=\frac{3x}{x^{2}(x - 8)}=\frac{3}{x(x - 8)}=\frac{3}{x^{2}-8x}\) for \(x
eq0\).

Step2: Find the limit as \(x\to\pm\infty\)

Use the rule for rational functions \(y = \frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}\). Here \(n = 0\) (degree of numerator) and \(m=2\) (degree of denominator).
We know that \(\lim_{x\to\pm\infty}\frac{3}{x^{2}-8x}\). Divide numerator and denominator by \(x^{2}\):
\(\lim_{x\to\pm\infty}\frac{\frac{3}{x^{2}}}{1-\frac{8}{x}}\)
As \(x\to\pm\infty\), \(\frac{3}{x^{2}}\to0\) and \(\frac{8}{x}\to0\). So \(\lim_{x\to\pm\infty}\frac{\frac{3}{x^{2}}}{1 - \frac{8}{x}}=0\)

Answer:

B. The function has one horizontal asymptote, \(y = 0\)