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 xaxis), 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 20032011 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 (straightline) 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.



Copyright © 20032012 Elizabeth Stapel  About  Terms of Use 




