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.