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The Purplemath Forums |
Asymptotes:
A Note Regarding 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:
...you can't do any long division, so you get:
So the horizontal asymptote is y = 0 (the x-axis), as you can see below:
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
...you can only do one trivial step in the division:
...which gives you:
So the horizontal asymptote is y = 2, as you
can see below:
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:
...you do the long division:
...and get:
So the slant (not horizontal) asymptote is y = 2x – 2, as you can see below:
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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