Infinity to the Zero Power

Date: 04/28/2001 at 04:16:17
From: Fady
Subject: Property of Numbers

Does (infinity)^zero = 1? Why ?

Date: 04/28/2001 at 05:48:18
From: Doctor Mitteldorf
Subject: Re: Property of Numbers

Dear Fady,

This isn't always true. The expression (infinity)^zero is really 
undefined, and can come out anything. But the way this expression 
comes up in practice is always the limit as something goes TOWARD 
infinity or TOWARD zero, or both.

For example, take x to the 0 power, and let x get bigger and bigger. 
Then the answer each time is 1, so the limit is also 1.

But take infinity to the y power, and let y get smaller and smaller. 
Then the answer is infinity each time, so the limit is also infinity.

The interesting case is where x and y are related in some way, so x 
gets bigger as y gets smaller in a coordinated way. Depending on the 
algebraic relation between x and y, the answer can come out 1 or 
infinity or anything in between.

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