Does the Lambert W function tend to infinity? Does $W(x)\to\infty$ as the real number $x\to+\infty$?
I find the equation (4.19) in paper https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf. It shows $$W(x)=\log x-\log\log x+\cdots$$.
Assuming $x\in\mathbb{R}^+$, $x\to+\infty$, is the following equation $$\frac{1}{1+W(x)}\to0$$ OK? Or it convergesto  a certain value? 

$W(\infty)=x\iff x\,e^x=\infty$ . Obviously, if x were finite, then so were $\infty$, which is absurd.
