This is an interesting discussion about why it isn't possible to define a "good" value for 0^0, i.e. a value which works in all cases, but the original riddle d

This is an interesting discussion about why it isn't possible to define a "good" value for 0^0, i.e. a value which works in all cases, but the original riddle doesn't really work. "Nice" operations like addition, multiplication and exponentiation all preserve continuity and the function we're given is discontinuous. The given solution contains a hidden special case, rolled into the author's proposed definition for x^0. From a formal mathematical standpoint, you define things in certain ways and then work with the definitions. It's not like there is a "correct" value for 0^0 out there that we can mount a search for. It's roughly the same with discussion of 0.999... - people still debate whether it's less than or equal to 1. You just have to use the definitions. In this case, 0.999... is shorthand for an infinite sum. An infinite sum is the limit of the sequence of its partial sums (where such exists). In this case the limit is provably one. So that's the value of 0.999... . If you look at a series like 1 - 1 + 1 - 1 + ..., its sequence of partial sums is divergent, so it doesn't have a value and that's that - although that doesn't mean we can't apply all sorts of alternative definitions to try to work with such a series, and some of them do in fact assign it a value. Such alternative definitions can produce really weird-looking results - one of the most striking is the result that (subject to certain definitions) 1 + 2 + 3 + 4 + ... is -1/12.