>>12 is an intuitive form, and doesn't fucking work at all. try explaining how exactly you can have sqrt2 apples from sqrt2(1 apple + 2 apples).
0^0 is indeterminate form but is usually taken to mean 1, since it makes binomial expansions nice, for one of many many other things.
>>17 is a nice non-rigorous demonstration of why 0/0 can be any number at all, since 0x = 0, for all real x.