Horizontal asymptotes are also referred to as end behavior. You are thinking about how the function behaves at extremely small numbers (-1,000, -10,000, -100,000...) and extremely large numbers (1,000, 10,000, 100,000...)
For each polynomial (the one in the numerator, and the one in the denominator), you can forget about every "lower degree" term, because it will be small in comparison to the first term. For your problem, 3x^3 will be WAY bigger than 28x^2 for large values of x. Similar thinking in the denominator simplifies your problem to just:
y= 3x^3/x^5
This can be further simplified to 3/x^2 . This is the only piece you have to consider for horizontal asymptotes. Now...if x gets quite large, this will clearly become extremely small, and approach 0. So your horizontal asymptote is at y=0.
You typically end up with one of three main cases:
1. Degree of numerator is HIGHER than degree of denominator, in which case there is no horizontal asymptote, and end behavior is either +/- infinity. (Note: if the degree is off by just one, you can have what is called a slant asymptote)
2. Degree of numerator EQUALS degree of denominator. In this case, the x parts cancel, and you are left with just the coefficients. If you have 10x^6/5x^6, your asymptote would just be 10/5, or y=2
3. Degree of numerator is LESS than degree of denominator (what you have). In this case, as explained above, the asymptote is y=0.
