Chapter 14, Problem 29RE

To determine

To find: The function $R(x)$ is undefined and find the asymptote it with

Explanation of Solution

Given information:

  $R(x)=\frac{x^3-2x^2+4x-8}{x^2-11x+18}$

  $R(x)=\frac{x^3-2x^2+4x-8}{x^2-11x+18}$

  $\begin{array}{l}=\frac{x^3-2x^2+4x-8}{(x-9)(x-2)}\\ =\frac{x^2(x-2)+4(x+2)}{(x-9)(x-2)}\\ =\frac{(x-2)(x^2+4)}{(x-9)(x-2)}\\ =\frac{(x^2+4)}{(x-9)}\end{array}$

Here, clearly for $x=9$ ,the

Function is undefined

And horizontal asymptote

Here, The vertical asymptote is

  $\begin{array}{l}x-9=0\\ x=9\end{array}$