where are you getting g(x) from? does $g(x)=x-\frac{f(x)}{f'(x)}$ for some $f(x)$ ?

$f(x)=e^{\frac{1}{2} \left(-x \left(\sqrt{x^2+1}+x\right)-\sinh ^{-1}(x)\right)}$ satisfies this.

Are you trying to find a root of this f(x)?

since $g(x)>x~~\forall x \in \Re$ it's pretty clear that g(x) won't converge when iterated.