Rational Functions

Question

Let r(x)=p(x)/q(x), where p and q are polynomials of degrees m and n respectively. What conditions on m and n ensure that the following statements are true?

a) lim r(x)= 0 x-->infinity

b) lim r(x)=k, with k cannot equal 0 x-->infinity

Answer 1

1. If m<n  then  lim r(x)= 0

 

2. If m=n  then  lim r(x)= k, where k is the ratio of the leading coefficient of p(x) and q(x), respectively.

 

For example, if  p(x)=2x^2+3x and q(x)=7x^2-2x+5, then k=2/7  so  

as x-->infinity lim(7x^2-2x+5)/(2x^2+3x) =2/7.

 

3. If m>n, then the limit is undefined (or approaches to + or - infinity)