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Re (tilly) 5: More Fun with Zero!by tilly (Archbishop) 
on Jul 24, 2001 at 06:41 UTC ( #99219=note: print w/ replies, xml )  Need Help?? 
Trust me. I said it correctly. 0^{0} has a
welldefined value. That value is 1. With the single
exception of the constructivists, I know of no branch of
mathematics where it makes any sense to say otherwise.
(And even the constructivists agree that that is the
answer when we are talking about integers.) The problem is that the function n^{m} goes insane near that point. If you are off by just a little bit, who knows where you wind up? What if you are dealing with a fractional power of a negative number? What if your exponent is positive but the base is exactly 0? What if the exponent is negative? No, for a floating point calculation, the issues with the continuity of the function become relevant. (And, of course, the constructivists admit of no noncontinuous functions of real numbers.) As for what you say about limits, my sincere advice is to forget that limits exist. Limits were not part of the mental framework that led Newton to create Calculus with his fluxions. (Which Bishop George Berkeley so rightly attacked.) Limits were not part of Leibniz' framework as he reassembled Calculus (not knowing what Newton had done) using a calculational aid called infinitesmals. (Which, contrary to popular mathematical belief, Berkeley spoke rather well of.) Limits were not part of Calculus over for centuries to come. Certainly Cauchy's attempt to make infinitesmals rigorous did not involve limits. Even though it led there (and led to the term, "Cauchy sequences"). And even after limits were invented in the 1870's, and slowly became part of the undergraduate curriculum, there is no evidence that they are the best  or even a good  way for students to understand Calculus. So forget limits. Calculus is not about formal manipulations involving artificial, unmotivated, computations leading to magic results that students blindly memorize. It is about the process of approximation. I firmly believe that you are better off if you clearly understand what, for instance, the derivative has to do with tangent lines than knowing an infinite number of useless tricks with limits. Before you are flabbergasted, you may enjoy reading what Knuth has to say on this topic. I don't agree with him on details, but I definitely agree that a solid understanding of approximation and errors will stand you in far better stead than confusing yourself with limits. (I speak, of course, as someone who is very familiar with limits...)
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