To determine the horizontal asymptote of the function

$f(x) = \frac{x^2 - 4}{3x^4 + 5},$

we analyze the degrees of the numerator and the denominator:

1. The degree of the numerator $x^2 - 4$ is $2$ (highest power of $x$ in the numerator).
2. The degree of the denominator $3x^4 + 5$ is $4$ (highest power of $x$ in the denominator).

Rule for Horizontal Asymptotes:

• If the degree of the numerator is less than the degree of the denominator ($\text{deg(numerator)} < \text{deg(denominator)}$), the horizontal asymptote is $y = 0$.
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.
• If the degree of the numerator is greater than the degree of the denominator ($\text{deg(numerator)} > \text{deg(denominator)}$), there is no horizontal asymptote.

Application:

Here, the degree of the numerator ($2$) is less than the degree of the denominator ($4$). Therefore, the horizontal asymptote is:

$y = 0$

Final Answer:

$\text{The horizontal asymptote is } y = 0.$

If you have further questions or need clarification, let me know!

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Five Related Questions:

1. How do you find vertical asymptotes of a rational function?
2. What happens when the degrees of the numerator and denominator are equal?
3. How are slant asymptotes different from horizontal asymptotes?
4. What is the role of the leading coefficients in determining horizontal asymptotes?
5. Can a rational function have both vertical and horizontal asymptotes?

Tip: For rational functions, always compare the degrees of the numerator and denominator to quickly determine the horizontal asymptote.