Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ F(x)=\frac{x^{4}-16}{x^{2}-2 x} $$

Short answer

Vertical asymptotes at \( x = 0 \) and \( x = 2 \). No horizontal or oblique asymptotes.

Step-by-step solution

- Factor the numerator and the denominator

Factor both the numerator and the denominator of the rational function, if possible. For this function: Numerator: \[ x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \]Denominator: \[ x^2 - 2x = x(x - 2) \]

- Identify vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at that same point). Set the denominator equal to zero and solve for x:\[ x(x - 2) = 0 \]Thus, the vertical asymptotes are at \( x = 0 \) and \( x = 2 \)

- Determine horizontal asymptotes

Consider the degree of the numerator and the denominator:- The degree of the numerator (\(x^4 - 16\)) is 4.- The degree of the denominator (\(x^2 - 2x\)) is 2.Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

- Determine oblique asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. Here, the numerator's degree (4) is two higher than the denominator's degree (2), so there is no oblique asymptote.

Key concepts

Vertical Asymptotes

Vertical asymptotes represent values where the function goes to positive or negative infinity due to the denominator approaching zero. For the function \( F(x)=\frac{x^{4}-16}{x^{2}-2 x} \), we start by factoring both the numerator and the denominator.

We factor the numerator as follows: \[ x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \] and the denominator as: \[ x^2 - 2x = x(x - 2) \].

Next, we find where the denominator is zero (and the numerator isn't zero at the same points). Setting up the equation and solving:

\[ x(x - 2) = 0 \] gives the solutions: \[ x = 0 \] and \[ x = 2 \]. Therefore, the vertical asymptotes are at:

• \[ x = 0 \]
• \[ x = 2 \]

Horizontal Asymptotes

Horizontal asymptotes illustrate the behavior of a function as it approaches very large positive or negative values of x. These are found by comparing the degrees of the numerator and the denominator of the rational function.

For \( F(x)=\frac{x^{4}-16}{x^{2}-2 x} \), note that:

• The degree of the numerator is 4.
• The degree of the denominator is 2.

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote for this function.

Oblique Asymptotes

Oblique or slant asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. They represent a diagonal line that the graph of the function approaches as \ x \ goes to positive or negative infinity.

In the given function \( F(x)=\frac{x^{4}-16}{x^{2}-2 x} \), the degree of the numerator (4) is two higher than the degree of the denominator (2).

Since the numerator's degree is not exactly one more than the denominator's degree, there are no oblique asymptotes for this function.