How to Find Vertical Asymptotes

Last reviewed October 15, 2026

A vertical asymptote is an x-value the graph runs up or down beside forever, without ever reaching. For a fraction, it comes from the denominator — but not every zero of the denominator makes one, and telling which do is most of the skill.

The method in three steps

1. Factor the numerator and the denominator completely.
2. Cancel any factor that appears in both. Each one that cancels completely is a hole, not an asymptote — note it and move on.
3. Set what is left of the denominator equal to zero and solve. Each solution is a vertical asymptote, written as the line x = a.

Step two is the one people skip, and it is the one examiners test. The rest of this page is mostly about why it matters.

Why the denominator

Dividing by a number close to zero gives a huge result: 1/0.1 is 10, 1/0.001 is 1000, 1/0.000001 is a million. If the bottom of a fraction heads towards zero while the top stays away from zero, the whole fraction grows without limit. That unbounded growth, as x approaches one particular value, is what a vertical asymptote is.

So vertical asymptotes live in the denominator. Zeros of the numerator do something quite different: they make the fraction zero, which puts the graph on the x-axis. Those are x-intercepts.

The phrase “while the top stays away from zero” is doing real work. If the top heads to zero at the same moment, the fraction is 0/0 in the limit, and that can settle on an ordinary number. That case is the hole, and it is coming up shortly.

A worked example

Find the vertical asymptotes of

y = (x + 1) / (x² − x − 6)

The denominator factors as (x − 3)(x + 2). The numerator is x + 1, which shares nothing with either, so nothing cancels. Setting the denominator to zero gives x = 3 and x = −2.

The trap: holes

Now find the vertical asymptotes of

y = (2x² − 8) / (x² + x − 6)

The tempting answer is to solve x² + x − 6 = 0, get x = −3 and x = 2, and stop. That is half wrong. Factor both halves first:

y = 2(x − 2)(x + 2) / ((x + 3)(x − 2))

The (x − 2) cancels, leaving (2x + 4)/(x + 3). That reduced expression is perfectly well behaved at x = 2: it equals 8/5. The original function is undefined there — its denominator really is zero — but it agrees with 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, not an asymptote.

The only vertical asymptote is x = −3.

The case that catches even careful students

In (x − 1) / (x − 1)², a factor of (x − 1) cancels — but one copy is left in the denominator, giving 1/(x − 1). So x = 1 is still a vertical asymptote. “Something cancelled” does not mean “hole”; what matters is whether the factor survives in the bottom.

Which way the graph goes

Knowing where the asymptotes are is half a sketch. The other half is knowing whether the curve shoots up or down on each side, and there is a quick way to tell.

Count how many times the factor appears. Near x = a, the factor (x − a) is a tiny number that changes sign as x passes through a. Raised to an odd power it still changes sign, so the function goes to +∞ on one side and −∞ on the other. Raised to an even power it is positive on both sides, so the function goes the same way on both.

Working out the signs by hand

For a single factor, test a value just either side. Take x = 3 in (x + 1)/((x − 3)(x + 2)):

• At x = 2.9: the top is 3.9 (positive), x − 3 is −0.1 (negative), x + 2 is 4.9 (positive). Positive over negative is negative, so the curve heads to −∞ on the left.
• At x = 3.1: only x − 3 has changed sign, so the fraction is positive and the curve heads to +∞ on the right.

You only need the signs, never the sizes, which is why a value a tenth either side is enough. The asymptote calculator lists both directions for every asymptote if you want to check.

Asymptotes that are not whole numbers

When the denominator does not factor nicely, solve it anyway. For (3x² + 1)/(2x² − 5), setting 2x² − 5 = 0 gives x² = 5/2, so

x = ±√(5/2) = ±√10 / 2 ≈ ±1.581

Give the exact form, x = ±√10/2, unless the question asks for decimals. For a quadratic denominator that does not factor, the quadratic formula does the same job.

When there are none

A function has no vertical asymptotes if its denominator, after cancelling, is never zero. Two common cases:

• The denominator has no real zeros. (x + 3)/(x² + 1) has none, because x² + 1 is at least 1 for every real x.
• Everything in the denominator cancels. (x² − 4)/(x − 2) reduces to x + 2, a straight line with a hole at (2, 4). It has no asymptotes of any kind.

Polynomials never have vertical asymptotes: they have no denominator to be zero.

tan, ln and other functions

The same idea — look for where the function divides by something tending to zero, or otherwise blows up — works beyond fractions of polynomials.

• tan x = sin x / cos x has a vertical asymptote wherever cos x = 0: at x = π/2 + kπ for every whole number k. So does sec x. cot x and csc x have them where sin x = 0, at x = kπ.
• ln x has a vertical asymptote at x = 0, approached from the right only: as x shrinks towards 0, ln x falls to −∞. It falls slowly — ln(0.000001) is only about −13.8 — but it never stops. In general ln(g(x)) has one wherever g(x) reaches zero, so ln(x − 3) has one at x = 3.
• Exponentials such as eˣ have no vertical asymptotes, but 1/(eˣ − 1) does, at x = 0, where its denominator is zero.

For these the asymptote calculator switches to a numerical search and confirms each candidate by checking that the function really does keep growing. It names the simple ones, so tan x comes back with asymptotes at ±π/2 and ±3π/2 rather than ±1.5708.

Mistakes that cost marks

1. Not cancelling first, and reporting a hole as an asymptote.
2. Calling it a hole because a factor cancelled, when a copy of that factor is still in the denominator.
3. Writing the asymptote as a number rather than a line. The answer is x = 3, not 3.
4. Looking at the numerator. Its zeros are intercepts, not asymptotes.
5. Drawing the curve joined across the asymptote. The branches on either side are separate pieces.

Questions

Find the asymptotes of any function