Infinity over Infinity

Date: 4/10/96 at 18:50:50
From: "Jeremy Vigneault"
Subject: Infinity question

Dear Dr. Math,

I have a question maybe you can answer.  My electronics teacher 
and I are having a disagreement.  He says that Infinity divided by 
Infinity equals one.  I most certainly disagree; I say that 
infinity over infinity is indeterminate, because infinity is a 
concept, not a finite number.  Could you please help me in this 
discrepancy?

Date: 4/12/96 at 21:36:13
From: Doctor Syd
Subject: Re: Infinity question

Dear Jeremy,

Good work!  You are correct.  Many people are confused by 
infinity; you are right that is a concept and not a number the way 
that 28 is a number.  There are sort of some different "sizes" of 
infinities, so this means that a quotient that looks like infinity 
over infinity can sometimes be  a real number, and sometimes it is 
just infinity.  

Maybe to prove your case to your teacher you could think about the 
following problem:

What is the limit as x approaches infinity of the expression 
x^2/(x-5)?   If you "plug in" infinity for x in this expression 
you getinfinity over infinity, but if you apply L'Hopital's Rule, 
you see the the numerator dominates the denominator (you can see 
this without L'Hopital's Rule, too:  x^2 gets big a lot faster 
than x does, right?), so the limit as x approaches infinity will 
be infinity, not one.  

Good question!  Hope this helped some.

-Doctor Syd,  The Math Forum