Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ P(x)=\frac{4 x^{2}}{x^{3}-1} $$

Short answer

Vertical asymptote: \(x = 1\), Horizontal asymptote: \(y = 0\), No oblique asymptotes.

Step-by-step solution

- Find Vertical Asymptotes

Find the values of x that make the denominator zero. Solve the equation: \(x^3 - 1 = 0\). Factorize it to get \( (x-1)(x^2+x+1) = 0 \). The solutions are \( x = 1 \) and \( x^2+x+1 = 0 \). Since the solutions to \( x^2+x+1 = 0 \) are complex, the only vertical asymptote is at \( x = 1 \).

- Determine Horizontal Asymptotes

Compare the degrees of the numerator and the denominator. The degree of the numerator \(4x^2\) is 2, and the degree of the denominator \(x^3-1\) is 3. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

- Check for Oblique Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Here, the degree of the numerator (2) is not one more than the degree of the denominator (3), so there are no oblique asymptotes.

Key concepts

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches as the input values get close to a certain number. To find the vertical asymptotes of a rational function, you need to set the denominator equal to zero and solve for the variable x. For our given function, \(P(x)=\frac{4 x^{2}}{x^{3}-1}\), we need to solve the equation \(x^3 - 1 = 0\).
Factorizing, we get \((x-1)(x^2+x+1) = 0\). Here, \(x = 1\) is our real solution. The equation \(x^2 + x + 1 = 0\) yields complex solutions, so it doesn't give us vertical asymptotes.
Therefore, the function has a vertical asymptote at \(x = 1\). The graph of \(P(x)\) will get infinitely close to this vertical line but will never touch or cross it.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values either go to positive or negative infinity. They appear as horizontal lines that the graph approaches but never touches. To determine horizontal asymptotes, we need to compare the degrees of the numerator and the denominator.
In \(P(x)=\frac{4 x^{2}}{x^{3}-1}\), the degree of the numerator is 2, while the degree of the denominator is 3. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always \(y = 0\).
This tells us that as x goes to positive or negative infinity, the value of \(P(x)\) will approach zero. The graph will get closer and closer to the x-axis but won't actually reach it.

Oblique Asymptotes

Oblique asymptotes, also known as slant asymptotes, occur when the degree of the numerator of a rational function is exactly one more than the degree of the denominator. They represent lines that the graph of the function will approach as x goes to positive or negative infinity.
Looking at \(P(x)=\frac{4 x^{2}}{x^{3}-1}\), the degree of the numerator is 2, and the degree of the denominator is 3. Since 2 is not one more than 3, there are no oblique asymptotes in this function.
Thus, you only need to check for oblique asymptotes when the degrees of the numerator and denominator meet this specific condition.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In the context of rational functions, understanding the degree of the numerator and the degree of the denominator helps us determine the presence of horizontal and oblique asymptotes.
For example, in the function \(P(x)=\frac{4 x^{2}}{x^{3}-1}\), the degree of \(4x^2\) is 2, and the degree of \(x^3 - 1\) is 3. This information was crucial for us to identify the horizontal asymptote at \(y = 0\) and to confirm that there are no oblique asymptotes.
Remember:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y=0\).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \(y = \frac{{a}}{{b}}\), where \(a\) and \(b\) are the leading coefficients.
• If the degree of the numerator is one more than the degree of the denominator, there's an oblique asymptote.