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### 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/
```
Associated Topics:
College Calculus
High School Calculus
High School Polynomials

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