If you see a dashed or dotted horizontal line on a graph, it refers to a horizontal asymptote (HA). In a rational function, an equation with a ratio of 2 polynomials, an asymptote is a line that curves closely toward the HA. The HA helps you see the end behavior of a rational function. In this article, we'll show you how to find the horizontal asymptote and interpret the results of your findings.

Things You Should Know

  • A horizontal asymptote is the dashed horizontal line on a graph. The graphed line of the function can approach or even cross the horizontal asymptote.
  • To find a horizontal asymptote, compare the degrees of the polynomials in the numerator and denominator of the rational function.
  • The degree of difference between the polynomials reveals where the horizontal asymptote sits on a graph.
Section 2 of 4:

How do I find a horizontal asymptote of a rational function?

  1. 1
    Remove all terms but the terms with the biggest exponents of . Since you're not actually solving an equation, you're simply comparing the leading terms in your rational function.[2]
    • For example, if your equation is , remove all but the leading terms to get .
    • As another example, your equation might be . After you remove all but the leading terms, you'll have .
    • To give you another example, if you have , ignore the constants to get .
  2. 2
    Simplify the ratio to find the horizontal asymptote for the function. Remember, you're not solving the equation—you're simplifying the leading terms in order to see the limits of the function.[3]
    • To go with the previous example of , cancel out both of the to get , which shows you the horizontal asymptote.
    • For the other previous example of , cancel out the top and take away from the denominator to get which becomes .
    • For the last example of , remove from the numerator and denominator to get .
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Section 3 of 4:

Horizontal Asymptote Rules and Results

  1. 1
    If the degree of the numerator and denominator is the same, use the coefficient ratio. If the polynomials in the numerator and denominator cancel each other out, you're left with the coefficients. Use the ratio of the coefficients to find the HA.[4]
    • Going back to our first example of , you ended up with . In this example, the HA is .
  2. 2
    If the numerator is at a lower degree, then the HA is located at y=0. This can be put another way—N<D means or the numerator is less than the denominator so .[5]
    • For our previous example of , you ended up with , so the horizontal asymptote is , which is also the x-axis.
  3. 3
    If the degree of the numerator is higher, there is no horizontal asymptote. It might help to remember this rule as N>D=no HA. When the numerator is greater than the denominator, it's not possible to have a horizontal asymptote.[6]
    • In the previous example that started with , you were left with . Since is larger than a nonexistent denominator, no HA is possible with this equation.
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About This Article

Jessica Gibson
Co-authored by:
wikiHow Staff Writer
This article was co-authored by wikiHow staff writer, Jessica Gibson. Jessica Gibson is a Writer and Editor who's been with wikiHow since 2014. After completing a year of art studies at the Emily Carr University in Vancouver, she graduated from Columbia College with a BA in History. Jessica also completed an MA in History from The University of Oregon in 2013. This article has been viewed 22,345 times.
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Co-authors: 4
Updated: October 25, 2022
Views: 22,345
Categories: Graphs
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