Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Testing for Horizontal Asymptotes

Date: 09/12/98 at 21:04:34
From: Kimberly Thomas
Subject: Horizontal asymptotes

My problem is to find asymptotes, the behavior of the function as x 
goes to +/- infinity, and the behavior of the function near any 
vertical asymptotes of (1-4x)/(2x+2). Then sketch a graph using the 

I understand the vertical asymptote (x when the denominator = 0, in 
this case -1). I'm stuck on the horizontal asymptote. I understand the 
definition, but I can't remember the exact formula for it.

Thanks for your help,

Date: 09/13/98 at 06:05:06
From: Doctor Pat
Subject: Re: Horizontal asymptotes


If the numerator and denominator are both polynomials (and yours are) 
you can use a sort of short version of l'Hopital's rule (which is a 
calculus theorem). The short rule, however, requires only a little 
memory and some thought. Look at the degree (highest power) of the 
numerator and denominator. There are three possible outcomes:  

   a) the highest power is in the numerator ---> the curve diverges to 
      plus/minus infinity and there is no horizontal asymptote

   b) the highest power is in the denominator ---> the curve converges  
      to zero as a horizontal asymptote
   c) both the numerator and the denominator have the same power ---> 
      the curve converges and has an asymptote at a value L where L is 
      the simplified ratio of the coefficients of the highest power 

Your problem is case c. Both the numerator and the denominator have 
degree one polynomials. The coefficient of that term in the numerator 
is -4 and in the denominator is 2, so the asymptote is -4/2 = -2.

Try these:

   (3x^2-5)/(4x+1) diverges to infinity (2nd power over first power)

   (4x+1)(3x^2-5) converges to zero (1st power over 2nd power)

   (3x^2 + 5x+4)/(5x^2-7x+2) converges to 3/5   

Hope this helps. Good luck.

- Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
Associated Topics:
High School Equations, Graphs, Translations

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994-2013 The Math Forum