Horizontal Asymptote

A horizontal asymptote is a horizontal line

y = L

that a graph gets closer and closer to as

x

heads toward positive or negative infinity. The function may never reach this line, or it may cross it at some points — but far out along the

x

-axis, the graph levels off near

y = L

A horizontal line $y = L$ is a horizontal asymptote of a function $f(x)$ if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ (or both). For a rational function $f(x) = \frac{P(x)}{Q(x)}$ , where $P$ and $Q$ are polynomials, the horizontal asymptote depends on the degrees of the numerator and denominator. A function can have at most two horizontal asymptotes — one as $x \to \infty$ and another as $x \to -\infty$ — though most common examples have the same asymptote in both directions.

Key Formula

\text{If } f(x) = \frac{a_n x^n + \cdots}{b_m x^m + \cdots}, \text{ then } y = \begin{cases} 0 & \text{if } n < m \\ \dfrac{a_n}{b_m} & \text{if } n = m \\ \text{none} & \text{if } n > m \end{cases}

Where:

$n$ = the degree of the numerator polynomial
$m$ = the degree of the denominator polynomial
$a_n$ = the leading coefficient of the numerator
$b_m$ = the leading coefficient of the denominator

Worked Example

Problem: Find the horizontal asymptote of

f(x) = \dfrac{6x^2 + 1}{3x^2 - 5}

Step 1: Identify the degree of the numerator and denominator.

\text{Numerator degree} = 2, \quad \text{Denominator degree} = 2

Step 2: Since the degrees are equal, divide the leading coefficients.

\frac{a_n}{b_m} = \frac{6}{3} = 2

Step 3: Verify by substituting a large value of

x

, such as

x = 1000

f(1000) = \frac{6(1000000) + 1}{3(1000000) - 5} = \frac{6000001}{2999995} \approx 2.0000

Answer: The horizontal asymptote is

y = 2

Visualization

Why It Matters

Horizontal asymptotes describe the long-run behavior of a function — what happens when inputs get extremely large or extremely small. In science and economics, this comes up constantly: a population model might level off at a carrying capacity, or a cost function might approach a minimum average cost. Recognizing horizontal asymptotes also helps you sketch accurate graphs of rational functions without plotting dozens of points.

Common Mistakes

Mistake: Believing a function can never cross its horizontal asymptote.

Correction: A function can cross a horizontal asymptote. For example,

f(x) = \frac{\sin x}{x}

crosses

y = 0

infinitely many times. The asymptote only describes behavior as

x \to \pm\infty

, not for all

x

values.

Mistake: Looking for a horizontal asymptote when the numerator's degree is greater than the denominator's.

Correction: When

n > m

, the function grows without bound and has no horizontal asymptote. If

n = m + 1

, the function has an oblique (slant) asymptote instead.

Related Terms

Asymptote — General term covering all types of asymptotes
Oblique Asymptote — Slant asymptote that occurs when degree of numerator is one more
End Behavior — Describes what a function does as x → ±∞