Intuitively you can think of exponentiation with an irrational exponent as a limit process like that.
2.
The limit process just described can be performed for any point in the domain of the squaring function.
3.
:An irrational power of a positive real number can be defined by a limit process, as you suggest.
4.
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
5.
With the advent of Fourier series, many analytical problems involving integrals came up whose satisfactory solution required interchanging limit processes and integral signs.
6.
Much of " WMCF " deals with the important concepts of infinity and of limit processes, seeking to explain how finite humans living in a finite world could ultimately conceive of the actual infinite.
7.
In calculus, arbitrary powers are usually instead defined in terms of exponentiation, so 2 ^ \ pi = \ exp ( \ pi \ cdot \ log 2 ), and exponentiation is defined as the inverse of the logarithm, where the log is defined as an integral ( a different sort of limit process ) : \ log x = \ int _ 1 ^ x { 1 \ over x } \, dx Preceding talk ) 17 : 34, 27 October 2009 ( UTC)
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