My girl friends's grandson asked me about 0^0, and it was difficult to give him as answer. He is only 9 years old, but knew that the zero power of positive numbers is one. He seemed to understand a simplified explanation based on a variable approaching a limit.

BTW: I can think of a good reason for e^0 = 1, but without more thought, I cannot come up with a good reason for x^0 = one for positive finite values of x.

The question has been rattling around in my head for a few days, and I wonder what some of you might think.

It seems to me that 0^0 is undefined in the same sense that 0 / 0 is undefined.

If you consider x^0 as positive x approaches zero, one looks like a good limit value. x^0 = 1 for all positive values of x, and I think it equals one for all negative values. Why should it not be one when x = 0?

If you consider 0^x as positive x approaches zero, zero looks like a good limit value. 0^x = zero for all positive nonzero values of x, why should it not be zero when x = 0?

I think that 0^x for negative x has infinity as a limit value, so there is a problem with 0^x as negative x approaches zero. Considering 0^x for negative x does not suggest zero or one for 0^0.

Does anybody have some thots on this?