Oblique Asymptote

Date: 4/13/96 at 23:46:53
From: Mohammad Mehdi Moshfeghian
Subject: Asymptotes

 Dear Dr Math.

I have a question about the oblique asymptote of the function
F( X ) = ( X^3 - 3X^2 + 2X - 8 ) / ( X^2 + 4X + 4 ).

According to the graph the line Y = X - 7 is the asymptote but it 
crosses the F ( X ) at about X = - 0. 769

I assumed that the asymptotes would not cross the functions.
I would appreciate any help with this problem.

  Regards,
    M.M.Moshfeghian

Date: 6/28/96 at 14:10:10
From: Doctor Jerry
Subject: Re: Asymptotes

If X^3 - 3X^2 + 2X - 8 is divided by X^2 + 4X + 4, the result is
( X^3 - 3X^2 + 2X - 8 ) / ( X^2 + 4X + 4 ) = X-7 + (26X+20)/(X^2+4X+4).

From this it is possible to see that the limit of F(X) as X 
increases without bound is X-7. If X = 10,000,000, for example, 
the fraction (26X+20)/(X^2+4X+4) is 2.6 times 10^(-6).  
So F(X) becomes and remains close to G(X) = X-7.

The important thing about one curve being an asymptote of another 
is how the curves behave as X increases without bound. Often, 
curves do not cross their asymptotes, but this is not a 
requirement.

-Doctor Jerry,  The Math Forum