A trivial solution of 
 $$1 = x^\frac{1}{x-a}$$  

 is x=1 as long as $$a \neq1$$

I would guess that behavior of $$\lim\_{a\to 1} x^\frac{1}{x-a}$$ 

is related to $$\lim\_{x\to 0} x^x$$

Interestingly, Wolframalpha list exactly five solutions to 

$$1 = x^\frac{1}{x-1}$$  

Each solution is in its $$x\approx\pm10^{20}\pm10^{20}{\times}i$$

Is there any meaning behind or is it just a bug?