> Playing around with a calculator, I found the following: > > If x=0 and y is a very small positive real number, then we have > x^y=0. Shifting x just slightly into the positive suddenly results in > x^y being very close to 1. > > Nothing earth-shattering here, but it seems to reinforce my > recommendation that 0^0 ought to be left undefined for any real-world > applications. f(x,y)=x^y simply behaves too strangely close to the > origin. > > Comments?
You've changed the question, haven't you? Didn't it start out as what is 0^0 in the natural numbers? Now it seems you're talking about reals (and even the real-world, shudder). For x^y in the reals you might want to look at this picture: https://en.wikipedia.org/wiki/File:X%5Ey.png. Note the text below it: "3D plot of z = abs(x)^y showing how different limits are reached along different curves approaching (0,0)."
-- Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton -- Geomancies