To solve this we can plot the function f(x)=x^x as x gets smaller in the positive range and see.
If we just take some random numbers:
x f(x)=x^x
1,00000 1,00000
0,90000 0,90953
0,80000 0,83651
0,70000 0,77906
0,60000 0,73602
0,50000 0,70711
0,40000 0,69314
0,30000 0,69685
0,20000 0,72478
0,10000 0,79433
It appears at first as if the function value decreases as x get smaller, but pretty soon it starts increasing again. This continues at ever decreasing values for x:
x f(x)=x^x
1,0E+00 1,000000000000000
1,0E-01 0,794328234724281
1,0E-02 0,954992586021436
1,0E-03 0,993116048420934
1,0E-04 0,999079389984462
1,0E-05 0,999884877372469
1,0E-06 0,999986184584876
1,0E-07 0,999998388191734
1,0E-08 0,999999815793210
1,0E-09 0,999999979276734
1,0E-10 0,999999997697415
1,0E-11 0,999999999746716
1,0E-12 0,999999999972369
1,0E-13 0,999999999997007
1,0E-14 0,999999999999678
1,0E-15 0,999999999999965
While 0^0 is undefined we can see that as the function f(x)=x^x approaches 1 as x approaches 0.
Mathematically this would be written as:
f(x)=x^x → 1 as lim x → 0
A plot I made in a hurry can be found here:
https://i.imgur.com/8IGYorr.png
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