Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ T(x)=\frac{x^{3}}{x^{4}-1} $$

Short answer

Vertical asymptotes at \(x = 1\) and \(x = -1\). Horizontal asymptote at \(y = 0\). No oblique asymptotes.

Step-by-step solution

- Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero. Set the denominator equal to zero and solve for x: \[ x^4 - 1 = 0 \]This can be factored as a difference of squares: \[ (x^2 - 1)(x^2 + 1) = 0 \]Further factoring gives: \[ (x - 1)(x + 1)(x^2 + 1) = 0 \]Solve each factor: \[ x - 1 = 0 \implies x = 1 \]\[ x + 1 = 0 \implies x = -1 \]\[ x^2 + 1 = 0 \implies x^2 = -1 \]Since \(x^2 = -1\) has no real solutions, the vertical asymptotes are at \(x = 1\) and \(x = -1\).

- Checking for Horizontal Asymptotes

The degrees of the numerator and the denominator determine the horizontal asymptotes. Here, the degree of the numerator (3) is less than the degree of the denominator (4). Thus, the horizontal asymptote is at:\[ y = 0 \]

- Checking for Oblique Asymptotes

Oblique asymptotes occur if the degree of the numerator is exactly one more than the degree of the denominator. Here, the degree of the numerator (3) is not greater by one compared to the degree of the denominator (4). Thus, there are no oblique asymptotes.

Key concepts

vertical asymptotes

Vertical asymptotes are important features of rational functions. They occur where the denominator of the function equals zero because division by zero is undefined. To find them in the function \( T(x)=\frac{x^{3}}{x^{4}-1} \), set the denominator equal to zero and solve for \( x \). In this example, we have:
\( x^4 - 1 = 0 \).
This equation can be factored as a difference of squares: \( (x^2 - 1)(x^2 + 1) = 0 \). Further factoring gives us: \( (x - 1)(x + 1)(x^2 + 1) = 0 \). By solving each factor separately, we find:

• \( x - 1 = 0 \rightarrow x = 1 \)
• \( x + 1 = 0 \rightarrow x = -1 \)

The factor \( x^2 + 1 = 0 \) leads to \( x^2 = -1 \), which has no real solutions. Therefore, the vertical asymptotes for this function are at \( x = 1 \) and \( x = -1 \). These values of \( x \) are points where the function goes to positive or negative infinity.

horizontal asymptotes

Horizontal asymptotes describe the behavior of a rational function as the input \( x \) approaches positive or negative infinity. They provide a boundary that the function values get closer to but never touch or cross. To determine horizontal asymptotes, we compare the degrees of the numerator and the denominator. For the function \( T(x)=\frac{x^{3}}{x^{4}-1} \), the degree of the numerator (3) is less than the degree of the denominator (4). This tells us that the horizontal asymptote is always at:
\( y = 0 \).
As \( x \) becomes very large or very small, the fraction \( \frac{x^{3}}{x^{4}-1} \) will tend towards zero because the denominator grows faster than the numerator. This insight into the behavior of the function helps us understand how it behaves at extreme values.

oblique asymptotes

Oblique asymptotes, also known as slant asymptotes, occur when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. These asymptotes reflect a linear trend that the function approaches but never touches. For the function \( T(x)=\frac{x^{3}}{x^{4}-1} \), the degree of the numerator (3) is not one more than the degree of the denominator (4):
In this case, since the degree of the numerator is less than the degree of the denominator, there are no oblique asymptotes.
Oblique asymptotes are unique and not common for every rational function. They only appear when specific degree conditions are met, which is not the case for this exercise.