Asymptote Calculator
There are three kinds, and the commonest mistake is writing down two that cannot both exist. A horizontal asymptote and a slant one never appear together — the degrees allow one or the other, never both.
This asymptote calculator works out all three for a rational function and shows the degree comparison that decides which you get, so the answer comes with the reason rather than on its own.
Vertical Asymptotes
These come from the bottom. Factor it, find where it is zero, and each of those values gives a vertical line the curve runs alongside without ever touching. That is all a vertical asymptote calculator has to do — until a factor cancels.
With one exception, and it is the one that costs marks. If the same factor also sits on top, it cancels — and then there is no asymptote there at all, only a hole: a single missing point on a curve that is otherwise perfectly ordinary. In (x²−4)/(x−2) the (x−2) cancels, so x = 2 is a hole and this function has no vertical asymptote. The calculator checks every factor and says which is which. (If you need the excluded values themselves rather than the asymptotes, the domain of a rational function calculator lists those.)
Horizontal and Slant Asymptotes — the Degrees Decide
Everything else comes from one comparison, and it is the whole job of a horizontal asymptote calculator. Look at the degree on top and the degree underneath:
Top lower. The fraction shrinks towards nothing as x grows, so the horizontal asymptote is y = 0. That is 1/(x²−4), for instance.
Degrees equal. The leading coefficients settle it. (3x²+1)/(x²−4) gives y = 3, because 3 divided by 1 is 3 — nothing else in either expression matters.
Top exactly one higher. Now there is no horizontal asymptote at all, and a slant one instead. Divide the top by the bottom: the quotient is the line. (x²+1)/(x−1) divides to give x + 1 with a remainder, and that remainder shrinks away as x grows — which is exactly why the curve settles onto y = x + 1.
Top more than one higher. Neither kind. The curve climbs too fast to settle onto any straight line.
Slant and oblique are two words for the same thing, so a slant asymptote calculator and an oblique asymptote calculator are looking for the same line. Whichever your class calls it, the method is the division above.
How to Use This Asymptote Calculator
Put the numerator in the top box and the denominator in the bottom, then press =. All three answers appear together, so it doubles as a vertical and horizontal asymptote calculator without your having to ask twice, with “none” written where there is none — because that is part of the answer, not a gap in it.
One tap opens the degree comparison with the reason spelled out, and under that the full asymptote calculator with steps — the working written out line by line. Everything stays exact: a horizontal asymptote at y = 1/2 is written as a fraction, never as 0.5. It is free to use online, with no account.
Frequently Asked Questions
How do you find a vertical asymptote?
How do you find a horizontal asymptote?
What is the difference between a slant and an oblique asymptote?
Can a function have both a horizontal and a slant asymptote?
Vertical asymptotes from the bottom, minus anything that cancels. Horizontal or slant from the degrees, never both. That is the whole topic.