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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.