Note: My current understanding is only at a college algebra level
From what I've seen online, in layman terms, the rules for horizontal asymptotes are as follows:
Rule 1) If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote will be $y=0$
Rule 2) If the numerator and denominator have equal degrees, then the horizontal asymptote will be a ratio of their leading coefficients
Rule 3) If the degree of the numerator is exactly one more than the degree of the denominator, then the oblique asymptote is found by dividing the numerator by the denominator. The resulting quotient is a linear expression which defines the oblique asymptote, and if there's a remainder, it's discarded.
For the first rule, I somewhat understand why the horizontal asymptote would be $y=0$
If the degree of the denominator is larger than the degree of the numerator, then the denominator is increasing at a faster rate than the numerator as $x\rightarrow\infty$. The numerator "can't keep up" and it would be getting divided by increasingly larger values so the outputs would be getting smaller and smaller approaching $0$.
Am I on the right track with my thinking here?
For rule 2 I'm not sure why the ratio of the leading coefficients of the numerator and denominator are used as the horizontal asymptote.
For Rule 3 why divide the numerator by the denominator to get an oblique asymptote? Why isn't there a horizontal asymptote instead?