"An asymptote to a curve is something like a tangent to the curve at infinity."
(1) This makes no sense to me because I am not familiar with "...a tangent to the curve at infinity."
(2) I posted THREE kinds of asymptotes, which means there must be three different answers needed, right?
An asymptote is a line that the graph of a function tries to reach or touch until close to infinity and yet the graph will never touch it. [I read somewhere long time ago that the graph might succeed at kissing and even crossing the horizontal asymptote. But that was long time ago and my mind is not good at recalling long time agos.) The distance between the asymptote and the graph of the function becomes smaller and smaller until no human-made microscope can even show the minute distances anymore but the gap is still there as the meaning of asymptote goes.
So we have a function y = f(x).
If y approaches a vertical line x = a and y just get higher and higher until it tries to reach positive infinity, or just get lower and lower until it tries to reach negative infinity, as x tries to approach the "a", then that vertical line x=a is an symptote of y=f(x). [So there is no y at f(a). There is no f(a). "a" is not a part of the domain of "f". Forget about these if they confuse you more.]
Note here that the x-distance, or the difference between the x-value and the next x-value, as x approaches the vertical asymptote, is getting smaller and smaller. [Eventually, that x-distance becomes so small that as if it amounts to zero already. Something related to finding the vertical asymptotes of a function--if there are--but ignore this also.]
If y approaches a horizontal line y = b and cannot reach that horizontal line no matter how far away x goes to the left or to the right, then that horizontal line is an asymptote of the y=f(x).
Note that here, the x-distance is getting bigger and bigger. The x is limited by how far it can go to the left or to the right, to negative infinty or to positive infinity horizontally. [Something related to finding the horizontal asymptotes of a function if there are. Umm, forget about this too.]
Now for the slant or oblique asymptote. This has the same characteristics as the horizontal asymptote in that the x goes to the right or to the left until it dies trying but, yet, still the y cannot kiss the oblique asymptote. The equation of the asymptote is a linear equation y = g(x), or y = bx +c.
[If in the function y = f(x) = h(x) /i(h), the numerator h(x) is one degree higher than the denominator i(x), then f(x) has an oblique asymptote. Yeah, forget about this too.]
[Asymptotes need not be straight lines only. Curved lines can be asymptotes too. If in the f(x) = h(x) /i(x), the numerator h(x) is two degrees higher than the denominator i(x), then the f(x) has a curved, a parabolic asymptote. Nahh, way out of topic, I know.]