Type a function to find all of its asymptotes. Rational functions are analyzed exactly: the denominator is factored, common factors become holes, and the degrees decide the horizontal or slant asymptote. Exponential, log, root and trig functions are handled with one-sided limits, and everything is drawn on the graph.
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Examples:
Rational functions are solved exactly. Also supported: sqrt, abs, ln, log, e^x, 2^x, sin, cos, tan and pi (these are analyzed with numeric limits).
Vertical asymptotes
Horizontal asymptotes
Slant (oblique) asymptote
Holes
Reasoning
Behavior near the vertical asymptotes
Graph
f(x)asymptoteshole
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How to use the Asymptote Calculator
Enter f(x), for example (2x^2 + 3x - 5)/(x^2 - 4).
Press Find asymptotes or pick an example.
Read the four result cards: vertical, horizontal, slant and holes.
Follow the reasoning and the left and right behavior table.
Check the graph: asymptotes are dashed, holes are open circles.
Rules for rational functions f(x) = P(x)/Q(x)
Degrees (n = deg P, m = deg Q)
End-behavior asymptote
Example
n < m
Horizontal y = 0
(3x + 1)/(x² + 1) → y = 0
n = m
Horizontal y = a/b (ratio of leading coefficients)
(2x² − 5)/(x² − 4) → y = 2
n = m + 1
Slant y = quotient of P ÷ Q
(x² + 1)/(x − 1) → y = x + 1
n ≥ m + 2
None (the graph follows a polynomial curve)
(x³ + 1)/(x − 1) behaves like x² + x + 1
Vertical asymptotes come from the zeros of Q that remain after cancelling common factors. A rational function can have any number of vertical asymptotes but at most one horizontal or slant asymptote.
Asymptotes of common functions
Function
Vertical
Horizontal
1/x
x = 0
y = 0
eˣ, bˣ (b > 1)
none
y = 0 as x → −∞
ln x, log x
x = 0 (from the right)
none
tan x
x = π/2 + kπ
none
arctan x
none
y = π/2 and y = −π/2
(2ˣ + 1)/(2ˣ − 1)
x = 0
y = 1 (x → +∞), y = −1 (x → −∞)
Tips
Always simplify first. Cancelling a common factor turns a would-be vertical asymptote into a hole.
Check both ends separately for non-rational functions; they can have different horizontal asymptotes on the left and right.
Use the signs near a vertical asymptote: if the factor appears to an even power, both sides go the same way; to an odd power they go opposite ways.
Slant asymptotes come from the same long division you use for improper fractions, so the quotient is the line and the remainder tells you where the graph crosses it.
Frequently asked questions
How do you find vertical asymptotes?
Simplify the function first, then find the x-values where the denominator is zero but the numerator is not. For a rational function these are the real zeros of the simplified denominator. If a factor cancels with the numerator, its zero is a hole instead of a vertical asymptote.
How do you find the horizontal asymptote of a rational function?
Compare the degrees n (numerator) and m (denominator). If n < m the asymptote is y = 0. If n = m it is y = (leading coefficient of the numerator)/(leading coefficient of the denominator). If n > m there is no horizontal asymptote.
When is there a slant asymptote?
A rational function has a slant (oblique) asymptote when the numerator's degree is exactly one more than the denominator's. Divide the polynomials: the quotient, a linear expression mx + b, is the asymptote, because the remainder fraction goes to 0 for large |x|.
Can a graph cross an asymptote?
A graph can never cross a vertical asymptote, because the function is undefined there. It can cross a horizontal or slant asymptote, sometimes several times, since those only describe what happens as x → ±∞. The calculator lists such crossing points for rational functions.
What is the difference between a hole and a vertical asymptote?
Both come from zeros of the denominator. If the zero cancels with a matching factor in the numerator, the function approaches a finite value and the graph has a hole (a single missing point). If it does not cancel, the function shoots off to ±∞ and there is a vertical asymptote.
Do exponential and log functions have asymptotes?
Yes. eˣ has the horizontal asymptote y = 0 as x → −∞, ln(x) has the vertical asymptote x = 0, and tan(x) has vertical asymptotes at x = π/2 + kπ. For such functions the calculator estimates the one-sided limits numerically and recognizes exact values where it can.
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Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.