One can enter solve sqrt(2)^x=x and obtain a result from W|A. It seems that W|A will not handle the more general case of a^x=x, though Mathematica is fine with it.

In[63]:= Solve[a^x == x, x]

During evaluation of In[63]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[63]= {{x -> -(ProductLog[-Log[a]]/Log[a])}}

This does not fully address the question since it is a multivalued result, so one must figure out which case is correct for the limit problem at hand.

Slightly off-question but I think this can help. I actually did some research on tetration (and extending it to complex realms) some weeks ago and I learnt the following through this Wolfram MathWorld page.

The value of the infinite power tower can be computed analytically by writing

Taking the logarithm of both sides and plugging back in to obtain

Solving for h(z) gives

.

Where W(z) is the Lambert W-function (Corless et al. 1996). h(z) converges iff e^(-e)<=x<=e^(1/e) (0.0659<=x<=1.4446; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35). This function is already in mathematica with the following name:

    ProductLog[x]

Just to prove it works:

I don't know what things you're working on but probably this method will be more efficient than solving for the limit manually. Hope it helped :)