02/26/2008, 09:31 PM

bo198214 Wrote:I dont understand why you invoke a polar graph here, the limits are very well defined and well-known:

You can already guess it when you look at the (cartesian) graph of the selfroot.

Here we know that for and hence for and hence . Finally again:

Hi,

I reread the beginning of the thread and it was dx^(1/dx) that was a problem, so lim x^1/x when x-> 0 is undefined in real numbers - that I understood from both jaydfox and Andy.

Since in my understanding infinity is 1/infinitesimal, x-> infinity is perhaps complementary to first one.

When You add another x making x^(1/x)^x You perhaps forcibly drive the expression into the limit You wish as exponent is higher level of infinitesimal than x is infinity. It does not prove (1/dx)^dx=1- these infinitesimals must be of same order.

I chose polar coordinates because there You can see what infinity of self root looks like - it is a unit circle-all 4 expressions tend to it, though from different sides. See attached file. As to self root close to 0, dx^1/dx, 2 of selfroots seem to start at 0, 2-at infinity, depending on combination of signs, so this is still undecided for me.