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Horizontal Asymptotes (page 2 of 4)

Sections: Vertical asymptotes, Horizontal asymptotes, Slant asymptotes, Examples


Whereas vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions. Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior far off to the sides of the graph. To get the idea of horizontal asymptotes, let's looks at some simple examples.

  • Find the horizontal asymptote of the following function:
    • y = [x + 2] / [x^2 + 1]

    The horizontal asymptote tells me, roughly, where the graph will go when x is really, really big. So I'll look at some very big values for x, some values of x very far from the origin:

      x y = [x + 2] / [x^2 + 1]
      100 000 0.0000099...
      10 000 0.0000999...
      1 000 0.0009979...
      100 0.0097990...
      10 0.0792079...
      1 0.5
      0 2
      1 1.5
      10 0.1188118...
      100 0.0101989...
      1 000 0.0010019...
      10 000 0.0001000...
      100 000 0.0000100...

    Off to the sides of the graph, where x is strongly negative (such as 1,000) or strongly positive (such as 10000), the "+2" and the "+1" in the expression for y really don't matter so much. I ended up having a really big number divided by a really big number squared, which "simplified" to be a very small number. The y-value came mostly from the "x" and the "x2". And since the x2 was "bigger" than the x, the x2 dragged the whole fraction down to y = 0 (that is, the x-axis) when x got big.

     
    I can see this behavior on the graph:

      

     graph of y = (x + 2)/(x^2 + 1)

    The graph shows some slightly interesting behavior in the middle, near the origin, but the rest of the graph is fairly boring, trailing along the x-axis.

    If I zoom in on the origin, I can also see that the graph crosses the horizontal asymptote (at the arrow):   Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

      graph of y = (x + 2)/(x^2 + 1)

    (It is common and perfectly okay to cross a horizontal asymptote. It's the vertical asymptotes that I'm not allowed to touch.)

    As I can see in the table of values and the graph, the horizontal asymptote is the x-axis.

      horizontal asymptote:  y = 0 (the x-axis)

In the above exercise, the degree on the denominator (namely, 2) was bigger than the degree on the numerator (namely, 1), and the horizontal asymptote was y = 0 (the x-axis). This property is always true: If the degree on x in the denominator is larger than the degree on x in the numerator, then the denominator, being "stronger", pulls the fraction down to the x-axis when x gets big. That is, if the polynomial in the denominator has a bigger leading exponent than the polynomial in the numerator, then the graph trails along the x-axis at the far right and the far left of the graph.


What happens if the degrees are the same in the numerator and denominator?

  • Find the horizontal asymptote of the following:
  •  

    ADVERTISEMENT

     

      y = [2x^2 - 11] / [x^2 + 9]

    Unlike the previous example, this function has degree-2 polynomials top and bottom; in particular, the degrees are the same in the numerator and the denominator. Since the degrees are the same, the numerator and denominator "pull" evenly; this graph should not drag down to the x-axis, nor should it shoot off to infinity. But where will it go?

    Again, I need to think in terms of big values for x. When x is really big, I'll have, roughly, twice something big (minus an eleven) divided by once something big (plus a nine). As you might guess from the last exercise, the "11" and the "+9" won't matter much for really big values of x. Far off to the sides of the graph, I'll roughly have "2x2/x2", which reduces to just 2. Does a table of values bear this out? Let's check:

      x y = [2x^2 - 11] / [x^2 + 9]
      100 000 1.9999999...
      10 000 1.9999997...
      1 000 1.9999710...
      100 1.9971026...
      10 1.7339449...
      1 0.9
      0 1.2222222...
      1 0.9
      10 1.7339449...
      100 1.9971026...
      1 000 1.9999710...
      10 000 1.9999997...
      100 000 1.9999999...

    For big values of x, the graph is, as expected, very close to y = 2.

     

    The graph reflects this:

      

     graph of y = [2x^2 - 11] / [x^2 + 9]

    Then my answer is: horizontal asymptote:  y = 2

In the example above, the degrees on the numerator and denominator were the same, and the horizontal asymptote turned out to be the horizontal line whose y-value was equal to the value found by dividing the leading coefficients of the two polynomials. This is always true: When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:

    y = (numerator's leading coefficient) / (denominator's leading coefficient)

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Cite this article as:

Stapel, Elizabeth. "Horizontal Asymptotes." Purplemath. Available from
    http://www.purplemath.com/modules/asymtote2.htm. Accessed
 

 



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