Asymptotes of Rational Functions: Vertical, Horizontal and Oblique
The asymptotes of a rational function describe its behaviour near the poles and at large |x|: vertical asymptotes at the zeros of the denominator, a horizontal asymptote when the degree of the numerator does not exceed the degree of the denominator, and an oblique (slant) asymptote when it is exactly one more. In brief: for f(x) = (2x2 + 1)/(x2 − 4), the vertical asymptotes are x = ±2 and the horizontal asymptote is y = 2. For g(x) = (x2 + 1)/(x − 1), the oblique asymptote is y = x + 1.
Asymptotes are lines that the graph approaches but, in general, does not reach (the horizontal and oblique ones can be crossed at finite x, but vertical ones never). They give the skeleton of the graph and are found without calculus by comparing degrees and using polynomial division. This guide explains the rules for each type, shows how to find them step by step, how to check where the graph crosses a horizontal asymptote and gives a table of the degree rules.
How do you find vertical asymptotes?
Write the function in lowest terms. Each zero a of the denominator that remains is a vertical asymptote x = a. Zeros that cancel give holes.
Example: f(x) = (x + 1)/(x2 − x − 6) = (x + 1)/((x − 3)(x + 2)). The vertical asymptotes are x = 3 and x = −2. For g(x) = (x − 3)/((x − 3)(x + 2)), the factor cancels, so only x = −2 is an asymptote, and there is a hole at x = 3.
How do you find horizontal asymptotes?
Compare the degree n of the numerator and the degree m of the denominator:
- n < m: the horizontal asymptote is y = 0.
- n = m: the horizontal asymptote is y = a/b, the ratio of the leading coefficients.
- n > m: there is no horizontal asymptote (the function grows without bound; if n = m + 1 there is an oblique asymptote).
The reason: for large |x|, only the highest powers matter, so f(x) ≈ a xn/(b xm) = (a/b) xn−m.
Example 1
(3x + 1)/(x2 + 2): n = 1 < m = 2, so y = 0. (2x2 + 1)/(x2 − 4): n = m = 2, so y = 2/1 = 2. (5x3)/(2x3 − x): y = 5/2.
Example 2: where does the graph cross the horizontal asymptote?
For f(x) = (2x2 + 1)/(x2 − 4) set f(x) = 2: 2x2 + 1 = 2x2 − 8, which gives 1 = −8, false. The graph never crosses y = 2. But h(x) = (x2 + x)/(x2 + 1) has the horizontal asymptote y = 1 and crosses it where x2 + x = x2 + 1, at x = 1. So a graph may cross a horizontal asymptote at finite x, though not for large |x|.
How do you find oblique asymptotes?
If the degree of the numerator is exactly one more than the degree of the denominator, divide the polynomials: f(x) = (mx + c) + R(x)/Q(x), where the remainder has a lower degree than the denominator. Then y = mx + c is the oblique asymptote, since the remainder fraction tends to 0.
Example 3: (x2 + 1)/(x − 1)
Divide: x2 + 1 = (x − 1)(x + 1) + 2. So f(x) = x + 1 + 2/(x − 1). The oblique asymptote is y = x + 1, and the vertical asymptote is x = 1. For large x, 2/(x − 1) is small and positive on the right and negative on the left, so the graph is above the line for x > 1 and below it for x < 1.
Example 4: (2x2 − 3x + 5)/(x + 2)
Division: 2x2 − 3x + 5 = (x + 2)(2x − 7) + 19. The oblique asymptote is y = 2x − 7, and the vertical asymptote is x = −2.
Example 5: no oblique asymptote
(x3 + 1)/(x − 1) has the numerator degree 3 and the denominator degree 1, a difference of 2. The quotient is a quadratic x2 + x + 1, so the graph follows a parabola. It has no horizontal or oblique asymptote (a parabolic asymptote only).
How do you use asymptotes to sketch a graph?
- Find the domain, holes and vertical asymptotes.
- Find the horizontal or oblique asymptote.
- Find the intercepts: f(0) and the zeros of the numerator.
- Use a sign chart to see where the function is positive or negative.
- Draw each branch between the asymptotes through the plotted points.
Example 6: f(x) = (x − 1)/(x2 − 4)
Vertical asymptotes: x = ±2. Horizontal asymptote: y = 0 (n < m). Intercepts: f(0) = −1/−4 = 1/4 and the zero x = 1. Signs: x < −2: (−)/(+) = −; −2 < x < 1: (−)/(−) = +; 1 < x < 2: (+)/(−) = −; x > 2: +. The graph crosses the x-axis at x = 1 and rises towards +∞ at −2+ and at 2+.
Degree rules at a glance
| Degrees | Asymptote at large |x| | Example | Result |
|---|---|---|---|
| n < m | y = 0 | (3x + 1)/(x2 + 2) | y = 0 |
| n = m | y = ratio of leading coefficients | (2x2 + 1)/(x2 − 4) | y = 2 |
| n = m + 1 | oblique line y = mx + c | (x2 + 1)/(x − 1) | y = x + 1 |
| n ≥ m + 2 | no line (polynomial of degree ≥ 2 behaviour) | (x3 + 1)/(x − 1) | follows x2 + x + 1 |
Where does the idea come from?
The word asymptote comes from the Greek for “not falling together”. Apollonius of Perga described the asymptotes of the hyperbola, the lines that the branches approach but never meet. The general study of asymptotes of curves was developed in the seventeenth and eighteenth centuries, and the notion of the limit made the concept precise in the nineteenth century.
Where are asymptotes used today?
Asymptotes show the long-term behaviour of a model: in the average cost C(n) = (500 + 3n)/n, the horizontal asymptote y = 3 is the lowest possible average cost per item. In pharmacology, a concentration curve approaches a steady level. In electronics, the gain of a filter tends to a constant at high frequencies, which is read from a horizontal asymptote of a transfer function.
Which mistakes are common?
Calling a hole an asymptote
Cancel common factors first. A cancelled factor gives a hole, not an asymptote.
Assuming the graph can never cross a horizontal asymptote
The graph can cross a horizontal asymptote at finite values of x. Only for large |x| does it approach it.
Using the leading terms for oblique asymptotes without dividing
The oblique asymptote needs the quotient of the division, including the constant term.
Forgetting that a rational function has at most one horizontal or oblique asymptote
Rational functions have the same limit at +∞ and −∞, so there is only one such line.