I have a couple quick questions about the exponential function handout. The first should be pretty easy: Rudin says that exp(0) = 1, using the definition given for exp(z). Does that mean that 0^0 = 1 and 0! = 1? That doesn't seem problematic, but I'm just double-checking that that's why exp(0) = 1.
The other question is probably a little more complicated. On p. 2, Rudin uses the definition of the exponential function to write the following equality:
[exp(z)][lim (exp(h) - 1)/h)] = exp z.
The limit is as h-->0. It would seem that Rudin's unwritten intermediate step is to show, from the definition of the exponential function, that
lim (exp(h) - 1)/h) = 1.
Can anyone give me a hint on how to see this? It looks sort of like the high school calculus problem that lim (sin x)/x is 1 as x goes to zero. In any case, I don't follow Rudin's calculation, so any help would be appreciated. Thanks, and happy 2000!