Write the equation of all horizontal asymptotes of the function f, of, x, equals, start fraction, 6, x, minus, 4, x, squared, divided by, 2, x, to the power 5 , plus, 3, x, to the power 4 , end fraction, .f(x)= 
2x 
5
 +3x 
4
 
6x−4x 
2
 
​
 .

To find the horizontal asymptotes of the function 

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

we need to analyze the behavior of the function as \(x\) approaches infinity (\(+\infty\)) and negative infinity (\(-\infty\)).

### Step 1: Determine the degrees of the numerator and denominator
- The degree of the numerator \(6x - 4x^2\) is 2 (the term with the highest power is \(-4x^2\)).
- The degree of the denominator \(2x^5 + 3x^4\) is 5 (the term with the highest power is \(2x^5\)).

### Step 2: Analyze the horizontal asymptote based on the degrees
- 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 equal to the degree of the denominator, the horizontal asymptote is \(y = \) the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (instead, there might be an oblique asymptote).

In this case, the degree of the numerator (2) is less than the degree of the denominator (5). Therefore, the horizontal asymptote is:

\[
y = 0
\]

### Final Answer
The equation of the horizontal asymptote is \(y = 0\).

Would you like any additional details or have any questions?

Here are 5 related questions to further explore:
1. How would the horizontal asymptote change if the degrees of the numerator and denominator were equal?
2. What are vertical asymptotes, and how can they be found for this function?
3. How would you find the slant (oblique) asymptotes if they exist?
4. How do horizontal asymptotes relate to the end behavior of the function?
5. What happens to the horizontal asymptote if you add a constant term to the function?

**Tip:** Always check the degrees of both the numerator and denominator first when determining horizontal asymptotes.

Write the equation of all horizontal asymptotes of the function f(x) = (6x - 4x^2) / (2x^5 + 3x^4).

Discover how to find the horizontal asymptotes of the rational function (6x - 4x^2) / (2x^5 + 3x^4). This problem involves key calculus concepts such as asymptotes and rational functions, making it ideal for students in Grades 11-12.

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