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 
information.

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,
Kim

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

Kim,

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 
      terms.  

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/