https://file.aiquickdraw.com/mathbot/file/1726243673362bwk4afeb.jpg

To determine which function has a horizontal asymptote at \( y = 0 \), we need to analyze each function's behavior as \( x \) approaches infinity.

### Steps:
1. **Horizontal asymptote determination**: A rational function \( f(x) = \frac{p(x)}{q(x)} \) will have a horizontal asymptote depending on the degrees of the numerator and denominator polynomials:
   - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
   - If the degrees are equal, the asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
   - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but possibly an oblique asymptote).

### Analyzing the Functions:
1. **\( h(x) = \frac{x^3 - 3}{x^4 + 4} \)**:  
   - The degree of the numerator is 3, and the degree of the denominator is 4.  
   - Since the numerator's degree is less than the denominator's degree, the horizontal asymptote is \( y = 0 \).

2. **\( h(x) = \frac{x^4 - 3}{x^2 + 4} \)**:  
   - The degree of the numerator is 4, and the degree of the denominator is 2.  
   - Since the numerator's degree is greater than the denominator's degree, there is no horizontal asymptote at \( y = 0 \).

3. **\( h(x) = \frac{x^5 - 3}{x^4 + 4} \)**:  
   - The degree of the numerator is 5, and the degree of the denominator is 4.  
   - Since the numerator's degree is greater than the denominator's degree, there is no horizontal asymptote at \( y = 0 \).

4. **\( h(x) = \frac{x^2 - 3}{x^3 + 4} \)** (Selected):  
   - The degree of the numerator is 2, and the degree of the denominator is 3.  
   - Since the numerator's degree is less than the denominator's degree, the horizontal asymptote is \( y = 0 \).

### Conclusion:
The correct answer is both **\( h(x) = \frac{x^3 - 3}{x^4 + 4} \)** and **\( h(x) = \frac{x^2 - 3}{x^3 + 4} \)**. However, only the last function has been selected in the image.

Would you like further clarification on asymptotes or the solution?

Here are 5 related questions:
1. How do you find the vertical asymptotes of a rational function?
2. What happens if the degrees of the numerator and denominator are equal?
3. Can a rational function have an oblique asymptote and how is it determined?
4. How do horizontal asymptotes relate to limits as \( x \to \infty \)?
5. What is the significance of the leading term in determining asymptotes?

**Tip:** Always compare the degrees of the numerator and denominator when determining the horizontal asymptote of a rational function.

Which of the following functions has a horizontal asymptote at y = 0?

Learn how to determine the horizontal asymptote of rational functions based on the degrees of the numerator and denominator. Suitable for high school students in Grades 10-12.

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