On 3/11/07, Andrey G. Grozin <A.G.Grozin at inp.nsk.su> wrote:
> lim x^y
>
> does not exist when both x and y tend to 0 (and are positive). The limit
> depends on the path along which (x,y) approaches (0,0). By choosing an
> appropriate path, one can easily obtain any non-negative value for this
> limit.
I agree, Andrey, but isn't the value limited to [0,1]?
Jay, consider a^b where a=exp(-1/x^k)) (k>0) and b=x; both a and b go
to 0 as x goes to 0.
This is equivalent to exp(-1/x^k)^x = exp(-x^(1-k)), and its limit as
x->0+ is 0 for k<1, 1/e for k=1, and 1 for k>1.
-s