How to Find Horizontal Asymptotes: Rules for Rational Functions

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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 1 of 4:

What is a horizontal asymptote?

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    When you look at a graph, the HA is the horizontal dashed or dotted line. When you plot the function, the graphed line might approach or cross the HA if it becomes infinitely large or infinitely small. [1]
    • Horizontal asymptotes can occur on both sides of the y-axis, so don't forget to look at both sides of your graph.
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Section 2 of 4:

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

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    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 .
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    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

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    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 .
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    This can be put another way—N<D means or the numerator is less than the denominator so .
    • For our previous example of , you ended up with , so the horizontal asymptote is , which is also the x-axis.
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    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. [5]
    • 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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Section 4 of 4:

Are horizontal asymptotes the same as slant asymptotes?

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    To find a slant asymptote , perform polynomial long division. Note that as you find the slant asymptote, you'll also find the vertical asymptote . [6]


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      References

      1. ↑ https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/2.10/primary/lesson/horizontal-asymptotes-pcalc/
      2. ↑ https://www.math.purdue.edu/academic/files/courses/2016summer/MA15800/Slantsymptotes.pdf
      3. ↑ https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/2.10/primary/lesson/horizontal-asymptotes-pcalc/
      4. ↑ https://youtu.be/wBZxVxiJS9I?t=289
      5. ↑ https://youtu.be/wBZxVxiJS9I?t=222
      6. ↑ https://www.math.purdue.edu/academic/files/courses/2016summer/MA15800/Slantsymptotes.pdf

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        Oct 10, 2023

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