Asymptote Calculator
Every vertical, horizontal and slant asymptote of a function, and every hole, found exactly from its factors — with the working, the direction the curve goes at each asymptote, and a graph that draws them the way a textbook does.
Solved exactly
A rational function, so every asymptote and hole below comes from its factors and degrees, with no rounding anywhere.
- Vertical asymptotesx = −2, x = 3
- Holesnone
- Horizontal asymptotey = 0
- Crosses it atx = −1
Which way it goes at each vertical asymptote
| Asymptote | Just to the left | Just to the right | Factor appears |
|---|---|---|---|
| x = −2 | f(x) → −∞ | f(x) → +∞ | once — opposite directions |
| x = 3 | f(x) → −∞ | f(x) → +∞ | once — opposite directions |
How they were found
- Write it as one fractionEverything is combined over a single denominator with whole-number coefficients, so the factors of the top and the bottom can be compared directly.
(x + 1) / (x² − x − 6) - Look for a common factorThe top and the bottom share no factor, so nothing cancels and there are no holes.
- Vertical asymptotes: zeros of what is left in the denominatorx² − x − 6 = 0 at x = −2, x = 3. The numerator is not zero at any of these after cancelling, so each one is a genuine vertical asymptote.
- Horizontal asymptote: compare the degreesThe degree of the top (1) is less than the degree of the bottom (2), so the function shrinks towards 0 at both ends. So y = 0.
- Does the graph cross its horizontal asymptote?The graph meets its horizontal asymptote wherever the remainder x + 1 is zero: at x = −1. A horizontal asymptote describes where the curve ends up, not a wall it cannot pass. Far to the left it approaches from below; far to the right, from above.
Holes first, then asymptotes
The mistake this calculator is built around: a zero in the denominator is not automatically a vertical asymptote. Take
y = (2x² − 8)/(x² + x − 6)
The denominator is zero at x = −3 and at x = 2. Most students, and a surprising number of online tools, report two vertical asymptotes. But factor both halves:
y = 2(x − 2)(x + 2) / ((x + 3)(x − 2))
The (x − 2) cancels. What is left is
(2x + 4)/(x + 3), which is perfectly well behaved at x = 2
— it equals 8/5 there. The original function is undefined at that
one point and identical to the reduced one everywhere else, so the graph
is a smooth curve with a single point missing at
(2, 8/5). That is a hole. The only vertical asymptote is
at x = −3.
So the calculator always cancels first. It finds the greatest common divisor of the top and the bottom — in exact fractions, so “is this a common factor” has a yes-or-no answer rather than one that depends on rounding — divides it out, and only then looks for asymptotes. Each cancelled factor becomes a hole, listed with its coordinates and drawn as a hollow circle on the graph.
There is one subtlety it also handles. In
(x − 1)/(x − 1)² a factor of
(x − 1) cancels, but one is left in the bottom, so x = 1
is still a vertical asymptote. Calling it a hole because a
factor cancelled is a second, rarer version of the same mistake.
Vertical asymptotes
After cancelling, every zero of what remains in the denominator is a
vertical asymptote. Irrational ones are given exactly:
(3x² + 1)/(2x² − 5) has asymptotes at
x = ±√10/2, not at ±1.5811.
For each one the calculator also says which way the curve goes on either
side. That comes from how many times the factor appears. A single factor,
like the (x + 3) above, changes sign as x passes through, so
the curve goes up on one side and down on the other. A squared factor,
like (x − 1)² in 1/(x − 1)², is
positive on both sides, so the curve goes the same way on both. Getting
this right is the difference between a sketch that earns the marks and
one that does not.
Horizontal, slant and polynomial asymptotes
What the function does far to the left and right comes from dividing the top by the bottom. The quotient is what the curve settles onto; the remainder fraction is what shrinks to nothing on the way.
| Degree of top vs bottom | The quotient is | So the asymptote is |
|---|---|---|
| Top lower | 0 | y = 0 |
| Equal | a constant: the ratio of the leading coefficients | that horizontal line |
| Top one higher | a linear expression | a slant (oblique) line |
| Top two or more higher | a polynomial of degree 2 or more | that curve — a polynomial asymptote |
The last row is the one almost every calculator gets lazy about. It is
true that (x³ + 1)/(x − 2) has no horizontal or slant
asymptote, and that is usually the end of the answer. But the curve does
not wander off at random: it hugs the parabola
y = x² + 2x + 4, which is the quotient, and the graph
here draws that parabola dashed so you can watch both branches settle onto
it.
Where the curve crosses its asymptote
A horizontal asymptote is not a wall. x/(x² + 1) passes
straight through its asymptote y = 0 at the origin, then approaches it
again from the other side. Students are often told this cannot happen; it
happens all the time.
The calculator finds every crossing exactly. Writing the function as quotient plus remainder over denominator, the curve is on its asymptote precisely where the remainder is zero. So the crossings are the zeros of the remainder — minus any that land on a pole or a hole. They are listed in the summary and marked on the graph.
Functions that are not fractions of polynomials
For ln x, tan x, eˣ, arctan x, and anything else outside the rational functions, there are no factors to cancel and no degrees to compare. The calculator switches to a numerical method and labels the result as such.
- Horizontal asymptotes are found at each end separately, as limits as x → −∞ and x → +∞. They can differ: arctan x approaches −π/2 on the left and π/2 on the right, and eˣ has an asymptote on the left only. A rational function can never do either.
- Vertical asymptotes are searched for in the window shown, and each candidate is confirmed by checking the function keeps growing as the distance shrinks — including slow growth like ln x near 0, and excluding functions that are merely steep. Values that are recognisably simple are named, so tan x gets asymptotes at ±π/2 and ±3π/2 rather than at ±1.5708.
What it does not do
- Asymptotes of curves that are not functions of x, such as the hyperbola x² − y² = 1. The conic sections calculator gives those.
- Vertical asymptotes of a non-rational function outside the search window. A rational function’s are all found, wherever they are.
- Asymptotes in polar or parametric form.
- Proofs of the limits in the numerical case. They are careful estimates, and labelled as estimates.
Questions
Why does it say "hole" where my textbook answer has an asymptote?
Because a factor that cancels from the top and the bottom makes a hole, not an asymptote — and this is the single most common mistake in the topic. (x² − 4)/(x − 2) has a zero denominator at x = 2, but (x − 2) cancels, leaving x + 2: the graph is a straight line with one point missing at (2, 4). There is nothing there for the curve to run off towards. The calculator cancels common factors exactly before it looks for asymptotes, and lists holes separately with their coordinates.
Can a graph cross its horizontal asymptote?
Yes, and the calculator tells you where. A horizontal asymptote describes where the curve ends up at the far left and far right, not a wall it cannot pass. x/(x² + 1) crosses its asymptote y = 0 at the origin and approaches it again from the other side. The crossings are the zeros of the remainder left over from dividing the top by the bottom, which is how they are computed here. Only vertical asymptotes can never be crossed.
What is a polynomial asymptote?
What a rational function settles onto when the top’s degree is two or more above the bottom’s. (x³ + 1)/(x − 2) has no horizontal or slant asymptote, but it is not shapeless far out: dividing gives x² + 2x + 4 plus a remainder that shrinks to nothing, so the curve hugs the parabola y = x² + 2x + 4. Most calculators report this case as "none". It is shown here, dashed on the graph, because it is the real answer.
Does it work for functions that are not fractions of polynomials?
Yes, numerically, and it says so. For ln x, tan x, eˣ or arctan x there are no factors to cancel, so horizontal asymptotes come from limits at each end separately — arctan approaches −π/2 on the left and π/2 on the right — and vertical asymptotes are searched for in a window you can change. Each candidate is confirmed by checking that the function keeps growing as the distance shrinks, so a function that is merely steep is not reported as having one. Simple values are named: tan x gets asymptotes at π/2 and 3π/2, not 1.5708.
Why does the curve go the same way on both sides of some asymptotes?
Because the factor is squared. Near x = 1, 1/(x − 1)² is a positive number divided by a tiny positive number on both sides, so it shoots upwards either way. 1/(x − 1) changes sign as x crosses 1, so it goes down on one side and up on the other. An odd power of the factor gives opposite directions, an even power the same direction, and the table under the result lists which for each asymptote.
Can I link to a particular function?
Yes. The function you type is kept in the address bar after a # sign, so copying the address shares the exact result. The guides on this site use this to open their worked examples directly in the calculator.