How to Find Horizontal Asymptotes
A horizontal asymptote is the height a graph levels off at as x runs to the far right or far left. For a fraction of polynomials it comes from comparing degrees, and three rules cover every case. For everything else it comes from a limit, and that is where the interesting exceptions live.
The three rules
For a rational function, compare the degree of the numerator (the highest power on top) with the degree of the denominator (the highest power on the bottom).
| Degrees | Horizontal asymptote | Memory aid |
|---|---|---|
| Top smaller | y = 0 | Bigger On Bottom: zero |
| Equal | y = (leading coefficient on top) / (leading coefficient on bottom) | Exponents Are The Same: Divide Coefficients |
| Top bigger | none | Bigger On Top: None |
The mnemonic is usually written BOBO BOTN EATS DC. It is worth having, as long as you also know why it works — that is what lets you deal with the functions it says nothing about.
Why the rules work
Far from the origin, the highest-power terms swamp everything else. When x is
a million, x² is a trillion, and a stray + 1 or
− 5 is irrelevant. So divide every term, top and bottom,
by the highest power of x in the denominator and watch what survives. For
(3x² + 1) / (2x² − 5)
divide everything by x²:
(3 + 1/x²) / (2 − 5/x²)
As x grows, 1/x² and 5/x² shrink to zero, leaving 3/2. That is the asymptote, y = 3/2. You can watch it happen: the function is about 1.544 at x = 10, 1.5004 at x = 100, and 1.500004 at x = 1000.
The same division explains the other two rules.
- Top smaller: after dividing by the bottom’s highest power, every term on top has a leftover power of x in its denominator, so the whole top shrinks to zero while the bottom tends to its leading coefficient. The fraction tends to 0.
- Top bigger: at least one term on top is left with a positive power of x, so the top grows without limit while the bottom settles. The fraction does not level off at all.
One example of each
-
(x + 1)/(x² − x − 6): degree 1 over degree 2, so y = 0. -
(3x² + 1)/(2x² − 5): degree 2 over degree 2, so y = 3/2. -
(x² + 1)/(x − 2): degree 2 over degree 1, so no horizontal asymptote.
Crossing the asymptote
Students are often told a graph never touches its horizontal asymptote. That is true of vertical asymptotes and false of horizontal ones. A horizontal asymptote is only about the far ends; in the middle of the graph the curve can cross it as often as it likes.
To find where, set the function equal to the asymptote’s value and
solve. For (2x² + x)/(x² + 1) the degrees are equal, so the
asymptote is y = 2/1 = 2. Then
(2x² + x)/(x² + 1) = 2
2x² + x = 2x² + 2
x = 2
So the graph crosses its own asymptote at (2, 2), and then approaches it from the other side.
A neater way to see it: dividing gives
2 + (x − 2)/(x² + 1). The curve is on its asymptote
exactly where the leftover fraction is zero, which is where its top,
x − 2, is zero. The
asymptote calculator
uses exactly this to find every crossing, and also says which side the curve
approaches from at each end.
When the top is bigger
“No horizontal asymptote” does not mean the curve wanders off at random. Long division tells you what it does.
- Top bigger by exactly one: the quotient is a straight line,
and that line is a slant (or oblique) asymptote.
(x² + 1)/(x − 2) = x + 2 + 5/(x − 2), so the curve approaches the line y = x + 2. - Top bigger by two or more: the quotient is a curve. For
(x³ + 1)/(x − 2)it is the parabola y = x² + 2x + 4, and the graph hugs it at both ends. This is sometimes called a polynomial or curvilinear asymptote.
A function cannot have both a horizontal and a slant asymptote: the degrees decide which one you get.
eˣ, ln x, arctan and roots
The degree rules only apply to fractions of polynomials. For anything else, go back to what a horizontal asymptote is: the limit of f(x) as x → +∞, and separately as x → −∞. Those two can be different, which a rational function never allows.
| Function | As x → −∞ | As x → +∞ |
|---|---|---|
eˣ | y = 0 | none |
e^(−x²) | y = 0 | y = 0 |
arctan x | y = −π/2 | y = π/2 |
x/√(x² + 1) | y = −1 | y = 1 |
1/(1 + e^(−x)) | y = 0 | y = 1 |
ln x | not defined | none |
Three of those deserve a word.
-
x/√(x² + 1): for large |x|, the root behaves like √(x²), which is |x|, not x. So the function behaves like x/|x|, which is 1 on the right and −1 on the left. Forgetting the absolute value gives the wrong answer on one side. -
ln xhas no horizontal asymptote. It grows very slowly — ln of a million is only about 13.8 — which makes it look as though it is levelling off, but it never stops: ln(10¹⁰⁰) is about 230. - The logistic curve
1/(1 + e^(−x)), used for population growth and in machine learning, has two horizontal asymptotes: it starts near 0 and finishes near 1.
Piecewise functions
Only the outermost pieces matter. Find the piece that applies as x → −∞ and take its limit there; then do the same for the piece that applies as x → +∞. Pieces defined on a bounded interval never reach the ends, so they cannot contribute.
For a function equal to eˣ for x < 0 and to
(2x + 1)/(x + 3) for x ≥ 0, the left end follows eˣ, giving
y = 0, and the right end follows the fraction, whose degrees
are equal, giving y = 2. Two different asymptotes, one at
each end.
Mistakes that cost marks
- Comparing the wrong terms. Use the highest powers, which
are not always written first: in
(1 + 3x²)/(5 − 2x²)the asymptote is 3/(−2) = −3/2, not 1/5. - Dropping a sign. The leading coefficient includes its sign, as that example shows.
- Saying a graph cannot cross its horizontal asymptote. It can, and questions are set to test exactly that.
- Writing “y = none”, or giving a horizontal asymptote when the top is bigger. Say there is none, and give the slant asymptote if the degree is one higher.
- Assuming both ends agree for a function that is not rational. Check each end separately.
Questions
Can a graph cross its horizontal asymptote?
Yes, as often as it likes. A horizontal asymptote only describes what the function approaches as x runs off to the far left or right; in between it can do anything. (2x² + x)/(x² + 1) crosses its asymptote y = 2 at x = 2, and sin(x)/x crosses y = 0 infinitely many times. To find the crossings, set the function equal to the asymptote value and solve.
Does ln x have a horizontal asymptote?
No. ln x keeps growing as x increases, just very slowly: ln(1,000,000) is about 13.8 and ln(10¹⁰⁰) is about 230. It never levels off at any value, so there is no horizontal asymptote on the right, and ln x does not exist for negative x, so there is nothing on the left either. It does have a vertical asymptote, at x = 0.
Are horizontal asymptotes discontinuities?
No. A horizontal asymptote is a statement about end behaviour — where the graph heads as x becomes very large or very negative — not a break in the graph. The function can be perfectly continuous everywhere and still have one, as arctan x does. It is vertical asymptotes that come with a discontinuity, an infinite one.
How do you find horizontal asymptotes of a piecewise function?
Only two pieces matter: the one that applies as x → −∞ and the one that applies as x → +∞. Find the limit of each at its own end. The pieces in the middle, on bounded intervals, cannot produce a horizontal asymptote, because they never reach the far ends. The two answers can differ, so a piecewise function can have one horizontal asymptote on the left and a different one on the right.
Can a function have two horizontal asymptotes?
Yes, one at each end — but never more than two, and never two for a rational function. arctan x approaches −π/2 on the left and π/2 on the right, and x/√(x² + 1) approaches −1 and 1. A rational function always approaches the same value at both ends, which is why its rules talk about "the" horizontal asymptote.
What does BOBO BOTN EATS DC mean?
A mnemonic for the three degree rules. Bigger On Bottom: the asymptote is y = 0. Bigger On Top: there is No horizontal asymptote. Exponents Are The Same: Divide the Coefficients of the leading terms. It is a fine memory aid, as long as you also know why the rules hold, which is what lets you handle the functions it does not cover.