One to the Power of InfinityDate: 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/ |
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