In response to the question Eddie Woo ended on, it has to do with the number that is being powered, and the power itself being equal in value, but different in growth metrics. This offers competing forces to contributing to the final value. On one hand, the number that you are taking the power of is getting smaller, on the other hand, you are increasing the number by taking it to a fractional power. This amounts to an initial loss to the value that results, followed by an eventual increase. A power and number being powered, that are both less than 1, can be displayed as a fraction, so you really also have a limit of (1/a)^(1/a) as a approaches infinity if you want to display as a fraction instead of a decimal. As you may see, from making a table, this means that (1/a) is being multiplied by itself less and less, causing less decay towards zero in the result (the value of the denominator of the denominator in the fraction gets bigger slower as the limit approaches infinity).
I haven’t proved it yet, and I’m just guessing, but I seem to think that you will find (0)^(0) is an irrational number when solving using limits, as long as you maintain that there is no such thing as nothing, and (0) is the smallest indivisible unit of mass, somehow because you would have to use infinite digits to display it, meaning it could not be presented as [fraction a/b with b not = 0.]. I tend to believe this because I am almost one hundred percent certain from studying lenses in Waves & Optics class that the smallest parts of the universe are only getting smaller, and doing so faster than we can measure with limited powers of observation.
I’ll end with a solved paradox: A man is crossing the street and gets halfway across. When he is halfway across, he continues crossing and covers half the remaining distance. If he continues to cover half the remaining distance, over and over again, he will never cross, but the gravel under his shoe would get finer and finer…
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