TITLE: How to define the $0^0$?
QUESTION [6 upvotes]: Possible Duplicate:
Zero to zero power 

According to Wolfram Alpha:

$0^0$ is indeterminate.

According to google:
$0^0=1$
According to my calculator: $0^0$ is undefined
Is there consensus regarding $0^0$? And what makes $0^0$ so problematic?

REPLY [4 votes]: $0^0=1$, and "$0^0$" is an indeterminate form.
The fact that it's a well defined expression in no way conflicts with the fact that it's an indeterminate form.
$0^0=1$ because it's an empty product.  Multiplying by no number is the same as multiplying by $1$; therefore when one multiplies by no number, the product is $1$.
It's indeterminate because one can let the pair $(x,y)$ approach $(0,0)$ along a path that makes the limit of $x^y$ equal to $5$ or to $1$ or to $\infty$, or to any of infinitely many other values.
If one approaches $(0,0)$ along any path that remains between two lines of positive slope, then the limit is $1$.
If $0^0$ were not equal to $1$, then the familiar expansion
$$
e^z= \frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{z!} + \cdots
$$
would fail when $z=0$, since the first term is $\dfrac{0^0}{0!}$.