Solving $e^\frac1x = x$ non-graphically? This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?). 
But how do you approach $e^\frac1x = x$ analytically for that solution? Is there a way?

Here are the steps
$$ \exp\left(\frac1x\right)=x $$
$$ \frac1x \exp\left(\frac1x\right)=1 $$
$$ \frac1x=W\left(1\right) =\Omega$$
$$ x=\frac1{\Omega}\approx\frac1{0.56714329}=1.76322$$
Where $W(z)$ is the Lambert W Function and $\Omega$ is the Omega Constant.
