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Zero to zero power

I'm wondering why $0^0$ is considered undefined. Why isn't 1 considered a valid solution?

Considering $0^0 = 1$ seems reasonable to me for two reasons:

  1. $\lim_{x \rightarrow 0} x^x = 1$

  2. $a^x$ would be a continuous function

Could you please explain why 1 can't be a solution and maybe provide some examples that show why having $0^0$ undefined is useful?

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marked as duplicate by Asaf Karagila, Douglas S. Stones, 5PM, Brian M. Scott, Clayton Jan 22 '13 at 0:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
In the Linked segment (right of the page, below the advertising) on the page of Zero to zero power you can find a bunch of other relevant questions. –  Asaf Karagila Jan 22 '13 at 0:20
    
I asked this question some time ago: math.stackexchange.com/questions/259514/how-to-define-the-00 –  Kasper Jan 22 '13 at 0:21

1 Answer 1

up vote 2 down vote accepted

0Because as a function $f(x,y): R^2 \rightarrow R = x^y$ we have two different values moving toward $f(0,0) = 0^0$. In other words, $f(0^+,0) = 1$ and $f(0,0^+) = 0$.

But beware that there are some places in mathematics which by convention accept one of these values. For example in some parts of combinatorics we have $0^0 = 1$ to ease the definition of some functions.

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