Asymptote Calculator

Asymptote calculator that tells holes apart from asymptotes

Enter a rational function and get every vertical, horizontal and oblique asymptote at once, with the denominator's roots, the degree comparison and the long division all shown. Cancelled factors are reported as removable holes rather than counted as asymptotes.

Works alongside the graphing calculator, so you can find the asymptotes and then see them.

Vertical, horizontal and oblique asymptotes, with removable holes told apart from them

The three kinds

Vertical, horizontal, oblique, and the one that is none of them

A vertical asymptote is a value of x the function cannot reach because the denominator is zero there while the numerator is not. The graph shoots off to ±∞ on either side of it.

A horizontal asymptote describes the far ends of the graph: the value f(x) settles towards as x runs off to ±∞. A curve is allowed to cross a horizontal asymptote in the middle of the graph, which surprises people. The asymptote is a statement about the ends, not a barrier.

An oblique or slant asymptote appears when the numerator's degree is exactly one higher than the denominator's. Polynomial long division splits the function into a straight line plus a remainder that fades to zero, and that line is the asymptote.

The fourth case is the trap: a factor shared by the top and the bottom cancels, and the graph has a removable hole there rather than an asymptote. This calculator separates the two, because reporting a hole as a vertical asymptote is the single most common error on this topic.

Vertical

x = 4

The denominator of (2x+1)/(x−4) is zero at x = 4 and the numerator is 9, so the value is unbounded there.

Horizontal

y = 2

Same degree top and bottom, so the ends settle at the ratio of the leading coefficients, 2/1.

Oblique

y = x + 3

(x²+1)/(x−3) divides to x + 3 with remainder 10, and the remainder fades to nothing.

A hole, not an asymptote

x = 2

In (x²−4)/(x−2) the factor (x−2) cancels, so the graph has a single missing point, not a break.

Method

How to find every asymptote of a rational function

Factor the top and the bottom.

Everything that follows depends on knowing which factors are shared. If you cannot factor by eye, the factoring calculator will do it and name the pattern it used.

Cancel the shared factors, and note where you cancelled.

Each cancelled factor (x − a) produces a hole at x = a, not an asymptote. The hole's y-value is what the simplified function gives at a.

Set the remaining denominator to zero.

Every solution is a vertical asymptote. If the denominator has no real roots, there are none, which is a perfectly normal answer.

Compare degrees for the end behaviour.

Bottom degree larger gives y = 0. Degrees equal gives y = the ratio of the leading coefficients. Top larger by exactly one gives an oblique asymptote. Top larger by two or more gives neither.

For an oblique asymptote, divide.

Polynomial long division gives a linear quotient plus a proper remainder. The quotient is the asymptote; the remainder tends to zero and can be discarded.

Worked examples

Three functions, all four cases between them

Vertical and horizontal

f(x) = (2x + 1)/(x − 4)

Nothing factors and nothing cancels, so there is no hole.
Set x − 4 = 0, giving a vertical asymptote at x = 4.
Top degree 1, bottom degree 1, so the degrees are equal.
Take the ratio of leading coefficients, 2/1, for a horizontal asymptote at y = 2.

x = 4 and y = 2

An oblique asymptote

f(x) = (x² + 1)/(x − 3)

x − 3 = 0 gives a vertical asymptote at x = 3.
Top degree 2, bottom degree 1, so the gap is exactly one and there is a slant asymptote rather than a horizontal one.
Divide: (x² + 1) ÷ (x − 3) = x + 3 remainder 10.
The remainder term 10/(x − 3) tends to 0 as x grows, so the curve approaches the line y = x + 3.

x = 3 and y = x + 3

A hole that looks like an asymptote

f(x) = (x² − 4)/(x − 2)

Factor the top: (x − 2)(x + 2).
The factor (x − 2) is shared, so it cancels and the function equals x + 2 everywhere except at x = 2.
There is therefore no vertical asymptote. There is a removable hole at x = 2, at height y = 4.
What remains is a straight line, which has no horizontal asymptote either.

No asymptotes. A hole at (2, 4).

Check your own function above →

Why it works

Degrees decide the ends

Divide every term of (3x² + 2x)/(x² − 1) by x² and you get (3 + 2/x)/(1 − 1/x²). As x grows, 2/x and 1/x² vanish, leaving 3/1. That is the whole argument behind the leading-coefficient rule, and it is worth doing once by hand so the rule stops being arbitrary.

The same division explains the other cases. If the bottom has the higher degree, dividing through leaves zero on top and a nonzero number underneath. If the top is higher, something is left over that keeps growing, and no horizontal line can describe it.

Related behaviour

Asymptotes are limits in disguise

A vertical asymptote at x = a is exactly the statement that the one-sided limits at a are infinite. A horizontal asymptote at y = L is the statement that lim(x→∞) f(x) = L. The two topics are the same mathematics with different vocabulary, which is why they are taught in the same week.

If you want the limits themselves rather than the lines, the limit calculator evaluates them and shows the degree comparison. To see the shape the asymptotes describe, plot the function in the graphing calculator.

Common questions

Asymptote calculator FAQ

How do you find the vertical asymptote of a function?

Factor the numerator and denominator and cancel anything they share. Then set the remaining denominator equal to zero. Every solution is a vertical asymptote. A factor that cancelled produces a hole at that x value instead, which is not an asymptote.

What is the difference between a hole and a vertical asymptote?

A hole appears where a factor cancels from both the top and the bottom, so the function is undefined at exactly one point but behaves normally around it. A vertical asymptote appears where the denominator is zero and the numerator is not, so the value runs to infinity. In (x² − 4)/(x − 2) the point x = 2 is a hole, not an asymptote.

How do you find a horizontal asymptote?

Compare the degrees. If the denominator's degree is larger, the horizontal asymptote is y = 0. If the degrees are equal, it is y = the ratio of the leading coefficients. If the numerator's degree is larger, there is no horizontal asymptote.

When does a function have a slant or oblique asymptote?

When the numerator's degree is exactly one more than the denominator's. Divide the numerator by the denominator; the linear quotient is the oblique asymptote and the remainder tends to zero. A function cannot have both a horizontal and an oblique asymptote.

Can a graph cross its horizontal asymptote?

Yes. A horizontal asymptote describes the behaviour at the far ends of the graph, not a line the curve is forbidden to touch. Many rational functions cross theirs once near the origin and then settle towards it. A vertical asymptote is different: the function can never cross one, because it is undefined there.

What if the denominator has no real roots?

Then there are no vertical asymptotes. 3x/(x² + 1) is a normal example: the denominator is never zero, so the graph is continuous everywhere, and only the horizontal asymptote y = 0 applies.

Does this asymptote calculator show the working?

Yes. It shows the roots of the denominator, which factors cancelled and why that makes a hole, the degree comparison, and the polynomial division when there is an oblique asymptote. It is free with no account.

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