Wondering if there was a way to prove this (or part of this) WITHOUT using calculus.

I'm sure we can at least show that f(k) is an increasing function, but I'm a bit'stuck. Tried various forms of induction, and it all turns out very messy. And of course, induction would only work for integers k.

Another thing I've been wondering, assumng we know f(k)->1 as k->0 and f(k)-> e as k-> inf, is this enough to show that f(k) is an increasing function for k>0? If so, why?

There is a way.
If you substitute 'inf' in the function you get the undefined form 1^(inf)

Use this, which I'm going to explain now, for calculating your limit: (1 + 1/k)^k = e^(k*ln(1+1/k)
And because your taking the power of the number 'e' this is a continuous function, so what you can do is calculate the limit of the power and that as a power of e will be equal to the limit of your function f(k). To calculate the limit of k*ln(1+1/k) you can use the method of L'Hopital.
So everything together gives you this,
-Rewrite the undefined form in a power of 'e'
Calculate the limit of the power of e with the rule of L'Hopital
The rule of L'Hopital requires knowledge of derivatives.
- you also have to remember that 1/(inf) = 0

That's all
I hope I was of an help to you,
Ariyan

I'm a bit puzzled here. When I took calculus, that limit was the definition of e.

(i) The definition of e is lim (1+1/n)^n
(ii) then, for any positive sequence xn->inf take the entire part of xn and use the "sandwich" trick
(iii) the same for any negative sequence
(iii) and then, any sequence (1+1/xn)^xn, xn->inf can be decomposed into a positive sequence as in (i) and a negative sequence as in (ii)