Chapter 14.3, Problem 79AYU

To determine

To find: The numbers at which $R$ is undefined and asymptotes or holes for $R$.

Answer to Problem 79AYU

Undefined for $x=-3$ and $x=2$.

Vertical asymptote: $x+3=0$; $x=2$.

Horizontal asymptote: $y=0$.

Explanation of Solution

Given:

$R\left(x\right)=\frac{x^3-2x^2+4x-8}{x^2+x-6}$

Calculation:

$R\left(x\right)=\frac{x^3-2x^2+4x-8}{x^2+x-6}$

$R\left(x\right)=\frac{\left(x^2+4\right)\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}$

The denominator of $R\left(x\right)$ becomes zero when

$\left(x+3\right)\left(x-2\right)=0$

$x=-3;x=2$

Thus $R$ is undefined for $x=-3$ and $x=2$.

This function has vertical asymptotes $x+3=0$ and $x=2$ and horizontal asymptote $y=0$.