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Asymptotes:
A Note Regarding If the degree is greater in the denominator, then this 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 2006-2008 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 one 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:
So, in a sense, 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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Copyright © 2006-2008 Elizabeth Stapel | About | Terms of Use |
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