Asymptote of a Function

Date: 06/02/2002 at 17:23:06
From: Sheryl
Subject: Finding the value of a when horizontal asymptote is given

I tried doing this question but I can't seem to figure it out.  I 
really hope you can help me:

Determine the value of a so that y = (ax + 5)/(3 - 6x) has a
horizontal asymptote at y = -2/3

Date: 06/02/2002 at 20:53:03
From: Doctor Ian
Subject: Re: Finding the value of a when horizontal asymptote is given

Hi Sheryl,

The thing about an asymptote is that when the value of x 
gets REALLY HUGE, the constant terms don't really contribute much 
to the value of the expression.  For example,

   a(1,000,000,000) + 5
   --------------------
   3 - 6(1,000,000,000)

is just about the same as

   a(1,000,000,000)
   -----------------
   -6(1,000,000,000)

which is just about the same as

    a
   --
   -6

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/02/2002 at 22:22:23
From: Sheryl
Subject: Finding the value of a when horizontal asymptote is given

I still can't figure out how to find a.

Date: 06/02/2002 at 22:29:08
From: Doctor Ian
Subject: Re: Finding the value of a when horizontal asymptote is given

Hi Sheryl,

In the function

         ax + 5
  f(x) = ------
         3 - 6x

when x gets really large, it's nearly the same as

          ax    a
  f(x) = --- = --
         -6x   -6

If the function is supposed to have an asymptote at 
f(x) = -2/3, the you need to find a value of a such that

    a   -2
   -- = --
   -6    3

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/02/2002 at 22:34:11
From: Sheryl
Subject: finding the value of a when horizontal asymptote is given

Yup!  Thank you so much!