How many horizontal asymptotes can the graph of a given rational function​ have? Give reasons for your answer.
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Part 1
Choose the correct choice below.
A.
The horizontal asymptotes vary in number depending on the degree of the numerator and denominator.​ Therefore, there can be infinitely many asymptotes.
B.
There is never any horizontal asymptote.
C.
There is always exactly one horizontal asymptote.
D.
If the numerator has larger degree than the​ denominator, there is no horizontal asymptote. If the numerator has smaller degree than the​ denominator, there is a horizontal asymptote at yequals0. If the numerator and denominator have the same​ degree, there is a horizontal asymptote at yequalsb where bnot equals0. ​Therefore, there is at most one horizontal asymptote.

The correct answer is:

**D**. If the numerator has a larger degree than the denominator, there is no horizontal asymptote. If the numerator has a smaller degree than the denominator, there is a horizontal asymptote at \( y = 0 \). If the numerator and denominator have the same degree, there is a horizontal asymptote at \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively. Therefore, there is at most one horizontal asymptote.

### Explanation:
A horizontal asymptote is related to the end behavior of the rational function as \( x \to \infty \) or \( x \to -\infty \). The number of horizontal asymptotes depends on the degrees of the numerator and denominator of the rational function:

1. **Degree of numerator < Degree of denominator**: The horizontal asymptote is \( y = 0 \).
2. **Degree of numerator = Degree of denominator**: The horizontal asymptote is \( y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} \).
3. **Degree of numerator > Degree of denominator**: There is no horizontal asymptote, though the graph may have an oblique (slant) asymptote.

Thus, a rational function can have **at most one horizontal asymptote**.

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Would you like more details or have any questions on this topic?

### Follow-up questions:
1. What happens if the numerator has a degree exactly one higher than the denominator?
2. How can you determine if a rational function has a slant asymptote?
3. How do vertical asymptotes differ from horizontal asymptotes in rational functions?
4. Can a rational function cross its horizontal asymptote?
5. How does the behavior of a rational function near infinity affect its horizontal asymptote?

**Tip:** Horizontal asymptotes describe end behavior, not necessarily the behavior of the graph near the origin or at specific points!

How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer.

Explore how to determine the number of horizontal asymptotes in a rational function based on the degrees of the numerator and denominator. Learn key concepts in rational functions, including asymptote behavior and conditions for horizontal asymptotes. Suitable for Grades 10-12.

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