One to the Power of Infinity

Date: 07/03/2001 at 02:18:51
From: Matt Boward
Subject: One to the infinite power

I recently had an instructor state that one to the infinite power does 
not equal one. That does not makes sense. If it is true, is there a 
relatively simple explanation?

Date: 07/03/2001 at 08:53:38
From: Doctor Peterson
Subject: Re: One to the infinite power

Hi, Matt.

This is one of several indeterminate forms; you can read about the 
concept in the Dr. Math FAQ:

   Dividing by 0
   http://mathforum.org/dr.math/faq/faq.divideby0.html   

You must first recognize that, since infinity is not really a number, 
an expression like this is defined only as a shorthand for a "limit"; 
this is a concept defined in calculus that allows us to let a variable 
approach some value and see what happens to the expression.

The idea is that you can make this equal different values by 
approaching 1^infinity in different ways. First, we know that 1 to any 
finite power is 1. But on the other hand, we know that any number 
other than one, raised to an infinite power, would be infinite if the 
base is greater than 1, and zero if the base were less than 1. It may 
seem natural to you to define

    1^infinity = lim[x->infinity] 1^x

which would be 1, since 1^x is always 1; but in order to have a solid 
definition, it turns out that we have to allow both numbers to vary, 
and define it as

    1^infinity = lim[x->1, y->infinity] x^y

But this is defined only if we get the same limiting value regardless 
of how we approach 1 and infinity. If we hold x constant at 1, we get 
1 as before; but if we hold y constant at infinity (which isn't really 
legal, but it gives us a good extreme case to picture), while x 
approaches 1 from above or from below, we get infinity or zero. If we 
let x and y simultaneously vary, we can get any answer we like, 
depending on which moves faster toward its destination. (This last 
phrase is an informal statement of L'Hopital's rule, which you will 
learn in calculus.)

There are explanations of this in our Dr. Math archives, which I found 
by searching for the phrase "1^infinity":

   What is 1^infinity?
   http://mathforum.org/dr.math/problems/hiscock12.10.98.html   

   Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms?
   http://mathforum.org/dr.math/problems/hanna5.8.98.html   

These give a little more detail.

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