A horizontal asymptote is a horizontal line y=cy=cy=c that the graph of a function gets closer and closer to as xxx goes to very large positive or negative values, even if it may cross that line for some finite xxx.

What is a horizontal asymptote?

  • A horizontal asymptote is a flat (horizontal) line, usually written as y=cy=cy=c, that describes the long‑term behavior of a function.
  • As x→∞x\to \infty x→∞ or x→−∞x\to -\infty x→−∞, the function values f(x)f(x)f(x) approach the constant ccc.
  • The line itself is not “part of” the graph; it is a guide to how the graph behaves far to the left and right.

Formally, y=Ly=Ly=L is a horizontal asymptote of f(x)f(x)f(x) if
lim⁡x→∞f(x)=L\lim_{x\to \infty}f(x)=Llimx→∞​f(x)=L or lim⁡x→−∞f(x)=L\lim_{x\to -\infty}f(x)=Llimx→−∞​f(x)=L.

Key facts (quick scoop style)

  • It describes end behavior, not what happens near the origin.
  • The graph can cross a horizontal asymptote at some points; the rule only talks about what happens as xxx goes to ±∞.
  • Common in rational functions f(x)=P(x)Q(x)f(x)=\dfrac{P(x)}{Q(x)}f(x)=Q(x)P(x)​, where PPP and QQQ are polynomials.

For rational functions: how to tell quickly

For f(x)=P(x)Q(x)f(x)=\dfrac{P(x)}{Q(x)}f(x)=Q(x)P(x)​, compare the degrees of the numerator and denominator.

[9][1][3][7] [1][3][7][9] [3][9][1]
Degree of numerator vs denominator Horizontal asymptote
Numerator degree < Denominator degree $$y = 0$$ (the x‑axis)
Numerator degree = Denominator degree $$y = \dfrac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$$
Numerator degree > Denominator degree No horizontal asymptote (you may get a slant/other type instead)

Tiny example story

Imagine driving on a road that starts on a hill and then flattens out, getting closer and closer to a perfectly flat highway at height 2.
The road is the graph of f(x)f(x)f(x); the perfectly flat highway at y=2y=2y=2 is the horizontal asymptote, because as you drive farther and farther, your height approaches 2 and stays near it.