Chapter 4.4, Problem 49AYU

In Problems 45-56, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

$T\left(x\right)\text{ = }\frac{x^3}{x^4\text{ - 1}}$

To determine

To find: Vertical, horizontal, or oblique asymptotes.

Answer to Problem 49AYU

Vertical asymptote are $x=-1$, 1, horizontal asymptote is $y=0$.

Explanation of Solution

Given:

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

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

$T$ is in lowest terms, and the zero of the denominator is $x^4-1=0$. The line $x=1$ and $x=-1$ are the vertical asymptote of the graph of $T$.

Therefore vertical asymptote are $x=-1$, 1.

Since the degree of the numerator, 3, is less than the degree of the denominator, 4, the degree of the numerator is less than the degree of the denominator; the line $y=0$is a horizontal asymptote.

Therefore horizontal asymptote is $y=0$.

Oblique asymptote: None.