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The problem you're working on asks to find the horizontal asymptote of the function:

\[
f(x) = \frac{5x^4 - 3x^3}{5x^6 - 2x}
\]

### Steps to solve:
1. **Compare the degrees** of the polynomials in the numerator and denominator.
   - The degree of the numerator \( 5x^4 - 3x^3 \) is 4 (the highest power of \( x \)).
   - The degree of the denominator \( 5x^6 - 2x \) is 6 (the highest power of \( x \)).

2. **Horizontal Asymptote Rules**:
   - If the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is \( y = 0 \).
   - If the degree of the numerator is **greater than** the degree of the denominator, there is no horizontal asymptote (it does not exist).
   - If the degrees are **equal**, the horizontal asymptote is the ratio of the leading coefficients.

In this case, the degree of the numerator is 4, and the degree of the denominator is 6. Since the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is:

\[
y = 0
\]

### Final Answer:
The horizontal asymptote is \( y = 0 \).

Would you like further details or have any questions? 

Here are 5 related questions you can explore:
1. What are vertical asymptotes, and how are they different from horizontal asymptotes?
2. How do you find oblique asymptotes when the degrees of the numerator and denominator differ by 1?
3. What happens if the numerator has a higher degree than the denominator?
4. How do limits help in identifying horizontal asymptotes?
5. What is the general process for analyzing the behavior of rational functions at infinity?

**Tip**: Always compare the degrees of the numerator and denominator first when determining horizontal asymptotes for rational functions.

Find the horizontal asymptote of \(f(x) = \frac{5x^4 - 3x^3}{5x^6 - 2x}\) if it has one (if it doesn’t, enter DNE).

Learn how to determine the horizontal asymptote of the function \(f(x) = \frac{5x^4 - 3x^3}{5x^6 - 2x}\) by comparing the degrees of the numerator and denominator. Suitable for advanced algebra students in Grades 11-12.

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