Similarly, if you're evaluating a limit and you get 0^0 you need to do more work. You can't just stop and say "oh, that's 1". It depends on what function you used to get there -- x^x will give you a different answer from ( e^(-1/x) )^x. Again, it has nothing to do with the continuity of exponentiation. The issue is whether the convention of 0^0=1 is correct in the specific part of mathematics you're working in.
The same argument can be made if you're working in the hyperreals, or if you're working with field axioms -- the convention 0^0 doesn't work in that context.
Please, by all means, use the convention 0^0=1 when it's appropriate. But understand that it's not always appropriate. Not every mathematician works in the particular subschool that you do; not every mathematician is going to find your convention appropriate.
What do you mean by "getting 0/0" in the process of evaluating limits?
The issue is not the continuity of x/x; the issue is whether setting a convention for 0/0 would give you the right value for a limit.
Please, tell me - what is the relation between lim f(a_n) and f(lim a_n) ?
From this point of view, the whole notion of "indeterminate form" makes just as little sense as distinguishing some arbitrary class of functions and calling them "elementary". Why are some points of discontinuity of some functions more special than other points of discontinuity of other functions? Why sin is more elementary than gamma? Historical heritage of confusion, I guess.