Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x(x - 1)^{2}}{(x + 3)^{3}} ).
determine the behavior of the graph on either side of any vertical asymptotes, if one exists. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
oa. it approaches ( infty ) on one side of the asymptote(s) at ( x=) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=).
(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)
ob. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=).
(type an integer or a simplified fraction. use a comma to separate answers as needed. type

Explanation:

Step1: Find the vertical asymptote

Set the denominator equal to zero: \((x + 3)^3=0\), so \(x=-3\) is the vertical asymptote.

Step2: Analyze the behavior near \(x = - 3\)

Let \(x=-3 + h\), where \(h\to0\). Then \(R(x)=\frac{(-3 + h)(-3 + h-1)^2}{(h)^3}=\frac{(-3 + h)(-4 + h)^2}{h^3}\).
Expand \((-4 + h)^2=16-8h+h^2\). So \(R(x)=\frac{(-3 + h)(16-8h+h^2)}{h^3}=\frac{-48 + 24h-3h^2+16h-8h^2+h^3}{h^3}=\frac{-48+40h - 11h^2+h^3}{h^3}\).
As \(h\to0^{+}\) (approaching \(x = - 3\) from the right), \(R(x)\approx\frac{-48}{h^3}\to-\infty\).
As \(h\to0^{-}\) (approaching \(x=-3\) from the left), \(R(x)\approx\frac{-48}{h^3}\to+\infty\) (since \(h^3\) is negative when \(h<0\)).

Answer:

A. It approaches \(\infty\) on one side of the asymptote(s) at \(x=-3\) and \(-\infty\) on the other.