Indeterminate Forms

Date: 09/18/97 at 13:24:37
From: White, Pam
Subject: Infinity

Dear Dr. Math,

We were sitting in Calculus class one day, when a question 
arose. What is infinity divided by infinity?  If you have an answer, 
we would appreciate an explanation also.  

Thank you for your time.
Beebe High School Calculus Class

Date: 09/18/97 at 15:21:58
From: Doctor Rob
Subject: Re: Infinity

This is an excellent question.

You cannot actually do arithmetic with "infinity" as if it were a 
number. You probably already know about this. Some operations can be 
defined quite well, but you run into trouble with others.

I guess you must mean by "infinity divided by infinity" something of 
the form lim(x->a)[f(x)/g(x)], where lim(x->a)[f(x)] = infinity and
lim(x->a)[g(x)] = infinity. As an example, you might have a = 0,
f(x) = 1/x^2, and g(x) = c/x^2. Then the quotient is a constant c, and 
the limit as x->0 is c. On the other hand, g(x) = 1/x is also 
possible, and the quotient is 1/x, whose limit as x->0 is infinity.  
Yet another possibility is g(x) = 1/x^3, so the quotient is x, whose 
limit as x->0 is 0.  Thus the limit of the quotient can be any real 
number or infinity.

The result of this situation is that it is very hard to say what 
"infinity divided by infinity" is. In fact, mathematicians call this 
one of the "indeterminate forms." There are seven of these. The others 
are "0/0", "0*infinity", "infinity - infinity", "1^infinity", "0^0", 
and "infinity^0".  For more about them, see:

  http://iq.orst.edu/mathsg/SandS/lHopital/index.html   

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/