Degree of a Rational Function

Date: 06/19/2003 at 08:28:35
From: Ban Nguyen
Subject: Rational function

f(x) = p(x)/q(x) 

You know that if the highest degree of p(x) is smaller than the 
highest degree of q(x), we have that the horizontal is y=0. Can you 
prove that?

Date: 06/19/2003 at 15:00:48
From: Doctor Rob
Subject: Re: Rational function

Thanks for writing to Ask Dr. Math.

Suppose that the degree of p is n and that of q is m.  Then write

           n                      n
   p(x) = SUM p[i]*x^(n-i) = x^n*SUM p[i]/x^i,
          i=0                    i=0

           m                      m
   q(x) = SUM q[i]*x^(m-i) = x^m*SUM q[i]/x^i.
          i=0                    i=0

Then

                                      n              m
    lim  f(x) =  lim  x^(n-m)* lim  {SUM p[i]/x^i}/{SUM q[i]/x^i}.
   x->oo        x->oo         x->oo  i=0            i=0

Now the last limit is easily seen to be p[n]/q[m], a finite nonzero
constant.  Since n < m, the preceding limit is easily seen to be 0.

Feel free to write again if I can help further.

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