What is the horizontal asymptote for the function y=x^2/(x^2-9)

To determine the horizontal asymptote for the function \( y = \frac{x^2}{x^2 - 9} \), we analyze the behavior of the function as \( x \to \infty \) and \( x \to -\infty \).

### Step 1: Analyzing the Degrees of the Numerator and Denominator
The degrees of the polynomial in both the numerator and the denominator are the same (both have degree 2).

### Step 2: Applying the Horizontal Asymptote Rule for Rational Functions
When the degrees of the numerator and the denominator are equal, the horizontal asymptote is found by dividing the leading coefficients of the highest degree terms in the numerator and denominator.

In this case:
\[
y = \frac{x^2}{x^2 - 9} = \frac{1}{1} = 1
\]

### Conclusion
The horizontal asymptote of \( y = \frac{x^2}{x^2 - 9} \) is:
\[
y = 1
\]

Would you like more details, or do you have any questions?

### Related Questions
1. What is the vertical asymptote for \( y = \frac{x^2}{x^2 - 9} \)?
2. How do you find oblique asymptotes for rational functions?
3. What happens to the graph of \( y = \frac{x^2}{x^2 - 9} \) near \( x = 3 \) and \( x = -3 \)?
4. How does the function \( y = \frac{x^2}{x^2 - 9} \) behave as \( x \to 0 \)?
5. How would the horizontal asymptote change if the numerator was \( x^3 \) instead of \( x^2 \)?

### Tip
For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always \( y = 0 \).

What is the horizontal asymptote for the function y=x^2/(x^2-9)?

Explore how to find the horizontal asymptote of the function y = x^2/(x^2 - 9) by analyzing the degrees of the numerator and denominator. This guide covers key concepts in rational functions and is suitable for high school math students in Grades 10-12.

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