what is a horizontal asymptote
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
limx→∞f(x)=L\lim_{x\to \infty}f(x)=Llimx→∞f(x)=L or limx→−∞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.
| Degree of numerator vs denominator | Horizontal asymptote |
|---|---|
| Numerator degree < Denominator degree | $$y = 0$$ (the x‑axis) | [9][1][3][7]
| Numerator degree = Denominator degree | $$y = \dfrac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$$ | [1][3][7][9]
| Numerator degree > Denominator degree | No horizontal asymptote (you may get a slant/other type instead) | [3][9][1]
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.