[spunkets]

"Limit x -->0 [a^x] = 1 "

If a = 0 there's no inverse. Exponentiation is defined as the inverse of the natural log.

a^x = exp(x/ln(a))

What's the ln of 0?

[Lonesome in Massachussets]

You might want to check your algebra. (Your point is still valid. )

a^x = exp(x*log_e(a))...

try it...

>> x = randn;a=randn; a^x - exp(x*log(a))

ans =

0

(Matlab is perfectly happy with complex arithmetic. 'log' means 'log_e', 'log10' is log_10. Excel doesn't do complex arithmetic, but 'LOG' means 'LOG10', unless you enter a second argument, specifying the radix. 'LN' is 'log_e' to Excel. Excel also thinks that 1900 was leap year and 2100 wouldn't be, making them inconsistent with any known calendar, except the Excel calendar.)

The meaning of "Limit x -->0 [f(x)] = 1 " is that you can name any number you want, except zero, call it 'eta', no matter how small you make eta, I name a value of x, 'epsilon', such that a^epsilon < 1+eta and that as eta gets smaller, epislon also gets smaller.