What it is
An asymptote is a straight line that a graph gets closer and closer to: the distance between the graph and the line tends to 0 as the graph heads off to infinity, sideways or up and down. The graph need not reach the line. The asymptote is not part of the graph and is often drawn dashed.
There are three kinds, by the direction of the line:
| Kind | Line | When it exists | Example |
|---|---|---|---|
| vertical | x = a | f(x) grows without bound, towards ∞ or −∞, as x approaches a | f(x) = 1/(x − 2) has x = 2 |
| horizontal | y = L | f(x) approaches the number L as x tends to ∞ or −∞ | f(x) = 1/x has y = 0 |
| oblique | y = kx + m with k ≠ 0 | f(x) − (kx + m) tends to 0 as x tends to ∞ or −∞ | f(x) = (2x² + x + 3)/(x − 1) has y = 2x + 3 |
Each kind has a page of its own with the method and more examples: vertical asymptote, horizontal asymptote and oblique asymptote.
A function can have several. f(x) = (3x + 1)/(x + 2) = 3 − 5/(x + 2) has two. The denominator is 0 at x = −2 while the numerator is 3 · (−2) + 1 = −5, so x = −2 is a vertical asymptote. The term 5/(x + 2) tends to 0 as x tends to ∞ or −∞, so y = 3 is a horizontal asymptote.
No asymptotes:
- f(x) = x². The function has a value for every x and grows without bound both ways, but no straight line gets closer and closer to the parabola.
- f(x) = sin x. The graph swings between −1 and 1 without approaching any line.
- f(x) = (x² − 4)/(x − 2). The denominator is 0 at x = 2, but for every other x the quotient is x + 2. The graph has a hole at (2, 4) and no asymptote.
Fractions are not the only functions with asymptotes. ln x has the vertical asymptote x = 0, eˣ has y = 0 as x tends to −∞, and tan x has the vertical asymptotes x = π/2 + nπ for every integer n.
Why it matters
Asymptotes show how a graph behaves where you cannot work out every point: near an x where the function has no value, and far out to the sides. They are also the frame when a graph is sketched by hand: mark the asymptotes, work out a few points, and draw the curve between them.
How to find them:
- Vertical asymptotes are where the denominator is 0 but the numerator is not, or where the function grows without bound in some other way. Check with a limit.
- Horizontal asymptotes come from the limits as x tends to ∞ and to −∞. Check both directions.
- Oblique asymptotes occur for a rational expression whose numerator has degree one higher than its denominator. Polynomial division gives (2x² + x + 3)/(x − 1) = 2x + 3 + 6/(x − 1), and since 6/(x − 1) tends to 0, the asymptote is y = 2x + 3.
Horizontal asymptotes appear in models where something approaches a limit. Bounded growth approaches the ceiling M, and y = M is a horizontal asymptote.
Plot f(x) = (3x + 1)/(x + 2) in the graphing calculator and watch the curve approach x = −2 and y = 3 without reaching them.
Common misconceptions
- “A graph can never cross its asymptote.” A graph can cross a horizontal or oblique asymptote. f(x) = x/(x² + 1) has y = 0 as a horizontal asymptote and still passes through (0, 0). At a vertical asymptote x = a, the values of the function tend to ∞ or −∞ as x approaches a.
- “The asymptote is part of the graph.” It is a separate line, and the graph only gets closer and closer to it.
- “Every graph has an asymptote.” Many have none, such as x² and sin x.
- “If the denominator is 0, there is a vertical asymptote.” Only if the numerator is not also 0 there. (x² − 4)/(x − 2) has a hole at (2, 4) and no asymptote.
- “Checking x → ∞ is enough.” To the left things can look different. eˣ has y = 0 as an asymptote to the left but no horizontal asymptote to the right.
Common questions
How many asymptotes can a function have?
Any number of vertical ones: tan x has infinitely many. To the right there is at most one horizontal or oblique
asymptote, and the same to the left, so at most two of those in all. arctan x has two horizontal ones, y = π/2 to
the right and y = −π/2 to the left.
What is the difference between an asymptote and a limit?
The limit is the number the function approaches, L in lim f(x) = L. The asymptote is the line y = L, which shows
the same thing on the graph.
Can an oblique and a horizontal asymptote exist on the same side?
No. On any one side there is either a horizontal or an oblique asymptote. An oblique line and a horizontal line
move apart far out, so the graph cannot get closer and closer to both.