A rational function can have more than one horizontal asymptote. True or False? Explain

False.

A rational function can have at most one horizontal asymptote. This is because the horizontal asymptote of a rational function \( \frac{P(x)}{Q(x)} \) (where \( P(x) \) and \( Q(x) \) are polynomials) is determined by the degrees of the polynomials in the numerator and the denominator.

Here's a summary of the rules for determining horizontal asymptotes for rational functions \( f(x) = \frac{P(x)}{Q(x)} \):

1. **If the degree of \( P(x) \) is less than the degree of \( Q(x) \)**, the horizontal asymptote is \( y = 0 \).
2. **If the degree of \( P(x) \) is equal to the degree of \( Q(x) \)**, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively.
3. **If the degree of \( P(x) \) is greater than the degree of \( Q(x) \)**, there is no horizontal asymptote. Instead, there may be an oblique asymptote.

Given these rules, a rational function cannot have more than one horizontal asymptote.

Would you like more details or have any other questions?

Here are 5 questions you can ask:

1. What are the steps to find the horizontal asymptote of a given rational function?
2. Can you explain the difference between horizontal and vertical asymptotes?
3. How do you determine the oblique asymptote of a rational function?
4. What happens to the rational function \( f(x) \) as \( x \) approaches infinity or negative infinity?
5. Can a rational function intersect its horizontal asymptote?

**Tip:** To determine the behavior of a rational function as \( x \) approaches infinity, compare the degrees of the numerator and the denominator.

Explore the concept of horizontal asymptotes in rational functions. Learn why a rational function can have at most one horizontal asymptote with detailed rules and examples. Suitable for high school students studying advanced algebra and calculus.

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