Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}}{x^{2}-x - 6} )
(type an ordered pair using integers or fractions.)

a. 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.)

b. it approaches ( infty ) on one side of the asymptote(s) at ( x=)
and ( -infty ) on the other.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

c. 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 each answer only once.)

d. there is no vertical asymptote

Explanation:

Step1: Find the vertical asymptotes

First, factor the denominator \(x^{2}-x - 6=(x - 3)(x+2)\). The vertical asymptotes occur at the values of \(x\) that make the denominator equal to zero. So, \(x = 3\) and \(x=-2\) are the vertical asymptotes.

Step2: Analyze the behavior near the vertical asymptotes

For \(x = 3\):
Let's consider the limit as \(x\to3^{+}\) (approaching \(3\) from the right). We can use test - values. Let \(x = 3.1\). Then \(R(x)=\frac{(3.1)^{2}}{(3.1 - 3)(3.1+2)}=\frac{9.61}{0.1\times5.1}\approx18.84\) (positive).
As \(x\to3^{-}\) (approaching \(3\) from the left), let \(x = 2.9\). Then \(R(x)=\frac{(2.9)^{2}}{(2.9 - 3)(2.9+2)}=\frac{8.41}{(- 0.1)\times4.9}\approx - 17.16\) (negative).
For \(x=-2\):
As \(x\to - 2^{+}\) (approaching \(-2\) from the right), let \(x=-1.9\). Then \(R(x)=\frac{(-1.9)^{2}}{(-1.9 - 3)(-1.9+2)}=\frac{3.61}{(-4.9)\times0.1}\approx - 7.37\) (negative).
As \(x\to - 2^{-}\) (approaching \(-2\) from the left), let \(x=-2.1\). Then \(R(x)=\frac{(-2.1)^{2}}{(-2.1 - 3)(-2.1+2)}=\frac{4.41}{(-5.1)\times(-0.1)}=\frac{4.41}{0.51}\approx8.65\) (positive).

Answer:

A. It approaches \(\infty\) on one side of the asymptote(s) at \(x=-2\) and \(-\infty\) on the other. It approaches either \(\infty\) or \(-\infty\) on both sides of the asymptote(s) at \(x = 3\)