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Asymptotes: A Note Regarding
     Horizontal and Slant Asymptotes

As an aside, you should note that all of the horizontal and slant asymptote rules pretty much reduce to the same thing: divide, and ignore the fractional part.


When the degree is greater in the denominator, then the polynomial fraction is like a proper fraction (such as 2/3 ) which cannot be converted to a mixed number other than trivially (as "0 2/3"). For instance, given:

    y = 2/(x + 1)

...you can't do any long division, so you get:

    y = 0 + 2/(x + 1)

So the horizontal asymptote is y = 0 (the x-axis), as you can see below:

    graph of y = 2/(x + 1)


If the degrees are the same, then the only division you can do is of the leading terms. For instance, given:   Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

 

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    y = 2x / (x + 1)

...you can only do one trivial step in the division:

    long division

...which gives you:

    y = 2 + (-2) / (x + 1)

So the horizontal asymptote is y = 2, as you can see below:

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


If the degree is higher on top, then the division gives a polynomial whose degree is the difference between the degrees of the numerator and denominator. Since you'll only be doing rationals where the numerator's degree is at most 1 greater than the denominator's degree, then the division will only give you, at most, a linear (straight-line) expression. For instance, given:

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

...you do the long division:

    long division

...and get:

    y = 2x - 2 + 2 / (x + 1)

So the slant (not horizontal) asymptote is y = 2x 2, as you can see below:

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


In a sense, then, you're always using long division to find the horizontal or slant asymptote. It's just that the long division is explicitly necessary only for the slant asymptote.

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Cite this article as:

Stapel, Elizabeth. "A Note Concerning Horizontal and Slant Asymptotes." Purplemath. Available from
    http://www.purplemath.com/modules/asymnote.htm. Accessed
 

 



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