Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{9}{(x - 5)(x^{2}-81)} ).
c. the function has no oblique asymptote.
determine the points, if any, at which the graph of ( r ) intersects the horizontal or oblique asymptote, if one exists. select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the graph of ( r ) intersects the horizontal or oblique asymptote at
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote.
c. there is no horizontal or oblique asymptote.

Explanation:

Step1: Find the horizontal asymptote

For a rational function \(R(x)=\frac{N(x)}{D(x)}\), where \(N(x)\) is the numerator and \(D(x)\) is the denominator. Here, \(N(x) = 9\) (degree \(n = 0\)) and \(D(x)=(x - 5)(x^{2}-81)=(x - 5)(x - 9)(x + 9)=x^{3}-5x^{2}-81x + 405\) (degree \(m=3\)).
When \(n

Step2: Set \(R(x)=0\)

Set \(\frac{9}{(x - 5)(x^{2}-81)}=0\).
Since the numerator \(9
eq0\) for all real - valued \(x\), the equation \(\frac{9}{(x - 5)(x^{2}-81)}=0\) has no solution.

Answer:

B. There is no point at which the graph of \(R\) intersects the horizontal or oblique asymptote.