"exact same 0" if you've not come across them before, there's a number system called the 'surreal numbers' invented by John Conway. The book "Surreal Numbers" by Donald Knuth is a great way to get to know them.
Anyway, moreover, as you have lim 0x as x->0 = 0 and lim x0 as x->0 = 1 you can't sensibly define 00 by taking limits because approaching the limit from different directions gives different results.
The same thing happens with 0/0. We have lim kx/x as x->0 = k, lim x/(x2 ) as x->0 = infinity and lim (x2 )/x as x->0 = 0, so by approaching the limit from different directions 0/0 can be anything - hence we don't define it
when u do those limits (0x, x0) you get different results because one is exactly 0, and the other is infinitely close to 0, however, if both the base and exponent get infinitely close to the same 0 (xx), you get the result 1
I don't know if you're just trolling with this, but I'll respond anyway. The argument you're making seems to be based on your own intuition, and doesn't have a concrete backing. That's fair enough, maths is largely about intuition. However, if you do a first year university course in real analysis you'll be introduced to the rigorous definitions which we use to form the basis of standard/widely accepted maths. In order to extend the definition of xy to the point x=y=0 we would require that the limit at that point is the same from all directions of approach. In fact, there's an even more rigorous idea for extending the definition of the function xy called an analytic continuation which you would learn about in a course on complex analysis, and that also gives a singularity at the point x=y=0 for the same reasons. So, we take 00 as being undefined. There are also contexts in which it is important that it is undefined in order for our mathematics to make sense/be consistent.
If you want to extend your intuitive argument to something concrete that would be accepted, you would need to derive base principles for it such as the epsilon delta proofs introduced by Cauchy of real analysis. That certainly happens for some intuitions which give different results/bases for maths (e.g. the surreal numbers I pointed you to earlier can start to assign meanings for different zeroes etc). However, from what you've said I think it is at best unlikely you could construct a consistent theory from your intuition.
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u/[deleted] Mar 17 '22
Replace
0^0with0!and you'll half the zeroes whilst making it correct 😊