tetration - inf superroot

Let's define the n'th iterative root ("rit") via

f(x,1) = x f(x,2) = x^x f(x,3) = x^(x^x)

such that

rit(y,3) = x --> f(x,3) = y

and consider the sequence

rit(3,1) , rit(3,2), rit(3,3),..., rit(3,k),... k=1,inf

what is x in

x = lim {k->inf} rit(3,k)

The sequence decreases from 3 down to e^(1/e) + eps
but I think, it cannot fall below.
k x
---------------------
1 3.000000 rit(3,k)
2 1.825455
4 1.563628
8 1.484080
16 1.457948
32 1.449171
64 1.446164
128 1.445135 rit(3,k)
....
->inf -> ?? rit(3,inf)
================================
compare other limits
inf 1.444668 =e^(1/e)
--------------------------------
inf 1.442250 =3^(1/3)

On the other hand, it should arrive at 3^(1/3)...
Do I actually overlook something and the sequence can
indeed cross e^(1/e)?

<urrks>

Gottfried
.

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