Paul Schlie <[EMAIL PROTECTED]> writes:
| > Gabriel Dos Reis wrote:
| > You probably noticed that in the polynomial expansion, you are using
| > an integer power -- which everybody agrees on yield 1 at the limit.
| >
| > I'm tlaking about 0^0, when you look at the limit of function x^y
|
| Out of curiosity, on what basis can one conclude:
|
| lim{|x|==|y|->0} x^y :: lim{|x|==|y|->0} (exp (* x (log y))) != 1 ?
The issue is not whether the limit of x^x, as x approaches 0, is 1 not.
We all, mathematically, agree on that.
The issue is whether the *bivariate* function x^y has a defined limit
at (0,0). And the answer is unambiguously no.
Checking just one path does NOT suffice to assrt that the limit
exists. (However, that might suffice to assert that a limit does not
exist).
I'm deeply burried somewhere in the middle-west deserts and I have no
much reliable connection, so I'll point you to the message
http://gcc.gnu.org/ml/gcc/2005-03/msg00469.html
where I've tried to taint this discussion with some realities from what
standard bodies think on the 0^0 arithmetic, and conterexample you can
check by yourself.
| As although it's logarithmic decomposition may yield intermediate complex
| values, and may diverge prior to converging as they approach their limit,
| it seems fairly obvious that the expression converges to the value of 1
You've transmuted the function x^y to the function x^x which is a
different beast. Existing of limit of the latter does not imply
existance of limit of the former. Again check the counterexamples in
the message I referred to above.
-- Gaby
