Chapter 14.3, Problem 77AYU

To determine

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

Answer to Problem 77AYU

Undefined for $x=1$ and $x=-\sqrt[3]{2}$.

Vertical asymptote: $x=1$; $x=-\sqrt[3]{2}$.

Horizontal asymptote: $y=0$.

Explanation of Solution

Given:

$R\left(x\right)=\frac{x^3-x^2+x-1}{x^4-x^3+2x-2}$

Calculation:

$R\left(x\right)=\frac{x^3-x^2+x-1}{x^4-x^3+2x-2}$

$R\left(x\right)=\frac{\left(x^2+1\right)\left(x-1\right)}{\left(x^3+2\right)\left(x-1\right)}$

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

$\left(x^3+2\right)\left(x-1\right)=0$

$x=-\sqrt[3]{2}$; $x=1$

Thus $R$ is undefined for $x=1$ and $x=-\sqrt[3]{2}$.

This function has vertical asymptotes $x=1$ and $x=-\sqrt[3]{2}$ and horizontal asymptote $y=0$.