An asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1x has both: x = 0 and y = 0.
Near x = 2 the denominator is close to 0, so the graph shoots up or down along the vertical asymptote. Far left and right, 1x − 2 is close to 0, so the graph levels off near y = 1.
Formal definition and notation
The line x = a is a vertical asymptote of f if f(x) → ∞ or f(x) → −∞ as x approaches a from the left or the right. The line y = b is a horizontal asymptote if limx → ∞f(x) = b or limx → −∞f(x) = b. A slant (oblique) asymptote y = mx + c with m ≠ 0 satisfies f(x) − (mx + c) → 0 as x → ±∞.
Symbol
Read it as
Meaning
x = a
x equals a
a vertical asymptote
y = b
y equals b
a horizontal asymptote
f(x) → ∞
f of x approaches infinity
the outputs grow without bound
limx → ∞f(x) = b
the limit of f of x as x approaches infinity is b
the horizontal asymptote is y = b
Key facts
A graph can never cross a vertical asymptote, because the function is undefined there.
A graph can cross a horizontal asymptote in the middle; the asymptote only describes the far ends.
A function has at most two horizontal asymptotes (one for each end) but can have many vertical ones.
Exponential functions have horizontal asymptotes: y = 2x + 3 approaches y = 3 on the left.
Logarithms have a vertical asymptote: y = logx has x = 0.
Asymptotes are found with limits; they describe end behavior and behavior near excluded values.
Horizontal asymptote rules for rational functions
Horizontal asymptote rules for rational functions
f(x) = p(x)q(x): compare degp and degq
degp < degq
horizontal asymptote y = 0
degp = degq
y = leading coefficient of pleading coefficient of q
degp = degq + 1
no horizontal asymptote; a slant asymptote from long division
q(a) = 0, p(a) ≠ 0
vertical asymptote x = a (after cancelling common factors)
If a factor cancels from the top and bottom, its zero gives a hole, not a vertical asymptote.
Try it: get close to x = 2
Interactive
The machine computes y = 1x − 2 + 1: subtract 2, take the reciprocal (power −1), add 1. It has run 3, 2.1 and 2.01, and the outputs explode near the vertical asymptote. Now try 100 and 1000: the outputs settle toward 1, the horizontal asymptote.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Asymptote examples, step by step
Example 1: Both asymptotes
Warm-up
Find the vertical and horizontal asymptotes of f(x) = 3x + 4.
The denominator is 0 at x = −4, and the numerator 3 is not 0 there: vertical asymptote x = −4.
The numerator has degree 0 and the denominator degree 1. Bottom-heavy means y = 0.
Answer: Vertical asymptote x = -4; horizontal asymptote y = 0.
Example 2: Equal degrees
Core
Find the asymptotes of g(x) = 2x + 1x − 3.
Denominator 0 at x = 3; the numerator is 7 ≠ 0 there, so x = 3 is a vertical asymptote.
A horizontal asymptote only describes the far left and right. In the middle, the graph may cross it.
✗ MistakeHorizontal asymptote of 4x22x2 + 1 is y = 0.
✓ InsteadIt is y = 42 = 2.
With equal degrees, the asymptote is the ratio of the leading coefficients. y = 0 is only for a bottom-heavy fraction.
Not to be confused with
Asymptotevs.Hole (removable discontinuity)
A hole is a single missing point where a factor cancels, like x = 2 in x − 2x − 2. At a vertical asymptote the graph shoots off to infinity instead.
Asymptotevs.Intercept
An intercept is a point where the graph meets an axis. An asymptote is a whole line the graph approaches. y = 1x + 1 has x-intercept−1 and horizontal asymptote y = 1.
Quick check: 3 questions
Question 1 of 3
What is the horizontal asymptote of f(x) = 6x2 − 13x2 + x?
Show answer and explanation
Answer: B. Equal degrees, so the asymptote is the ratio of leading coefficients: 63 = 2. y = 0 is for a bottom-heavy fraction, and −1 and 6 use the wrong coefficients.
Question 2 of 3
Which lines are the vertical asymptotes of g(x) = xx2 − 9?
Show answer and explanation
Answer: D.x2 − 9 = (x − 3)(x + 3) is 0 at 3 and −3, and the numerator is not 0 at either. 9 forgets to take the square root, x = 0 is where the numerator is 0 (an x-intercept), and x = 3 only misses the negative root.
Question 3 of 3
What does the graph of h(x) = x − 5x2 − 25 have at x = 5?
Show answer and explanation
Answer: A. The factor x − 5 cancels: h(x) = 1x + 5 for x ≠ 5. So there is a hole at x = 5, and the vertical asymptote is at x = −5. It is not an intercept because h(5) is undefined.
Asymptote: frequently asked questions
How do you find a vertical asymptote?
Factor the numerator and denominator, cancel any common factors, and then set the remaining denominator equal to 0. Each solution x = a is a vertical asymptote. For x + 1x2 − x − 6 = x + 1(x − 3)(x + 2), the vertical asymptotes are x = 3 and x = −2. Values from cancelled factors are holes instead.
How do you find a horizontal asymptote?
For a rational function, compare degrees. If the bottom degree is bigger, the asymptote is y = 0. If they are equal, divide the leading coefficients. If the top is bigger, there is no horizontal asymptote. For other functions, ask what f(x) approaches as x gets very large, which is the limit as x → ±∞.
Can a graph cross an asymptote?
It can cross a horizontal or slant asymptote, because those only describe the far ends of the graph. For example, y = xx2 + 1 passes through the origin, on its asymptote y = 0. It can never cross a vertical asymptote, because the function has no value at that x.
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