What is the limit of the sequence defined as$$(x)_{ n\geq 0},\qquad x_0 = 1,\quadx_{n+1} = x_{n} + \frac{2}{x_{n}}\ ?.$$Since it's an increasing sequence, I suppose it is also divergent with a limit of $\infty$, but I don't know how to rigorously prove it.
$$\mbox{Then, what is the limit of}\quad {x_{n} \over n^{1/2}}\ ?.$$
I don't have the intuition of how I should even approach this kind of limit problem. If you could give me some hints or point me out to resources I would much appreciate $\mbox{it !.}$