Why is $0^0$ undefined?

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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:

$\lim_{x \rightarrow 0} x^x = 1$
$a^x$ 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?

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 00:20
I asked this question some time ago: http://math.stackexchange.com/questions/259514/how-to-define-the-00 — Kasper, Jan 22 '13 at 00:21

score 3 · Accepted Answer · answered Jan 22 '13 at 00:24

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.

Why is $0^0$ undefined?

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