Trust me. I said it correctly. 0^{0} has a
well-defined 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 non-continuous
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...) | [reply] |