I was fiddling around with tetration and I stumbled across an interesting idea, ${^\infty}{\sqrt[x]{x}}$. I messed around with the concept a little bit and I had the following idea:
Let ${^\infty}y = x$, then $$ {^\infty}y = x = y^x $$ Now we have $$ x = y^x,\\ y = \sqrt[x]{x} $$ Thus $$ \sqrt[x]{x}^{\sqrt[x]{x}^{\sqrt[x]{x}^\cdots}} = x $$ I think it makes sense, but I'm not entirely sure if it's right or not. Plugging in some values for x and shoving it into a recursive function in python showed that it worked for $x = 2$, but at $x = 3$ it converged at $2.478051576300804$. I'm not sure if this is because my identity is false or because floating-point precision isn't precise enough. So is this statement true?
It is not only true, but mentioned explicitly in the Wikipedia article on the subject. :-) In fact, it has been known since the time of Euler. What you forgot to add is that infinite tetration only converges for $x\in\Big[e^{-e}~,~\sqrt[e]e\Big]$.