Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

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

Short answer

$y=0$ is the horizontal asymptote for given rational expression.

Step-by-step solution

Step 1. Given information

Given rational expression:

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

Step 2. Write the concept

If $n<m$(the degree of the numerator is less than the degree of the denominator), then R is a proper rational

function, and the graph of R will have the horizontal asymptote y = 0 (the x-axis).

Step 3. Calculation

Here, $n=3,m=4$

Now,

$$\begin{gathered} consider \\ P\left(x\right)=\frac{x^3}{x^4-1} \\ =li{m}_{x\to \infty }\frac{x^3}{x^4(1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x^4$}\right.})} \\ =0 \end{gathered}$$