What kind of growth does this function have? $f(n) = f(n-1) + \dfrac{1}{f(n-1)};~n\in\mathbb{N}$, where $f(0) = 1$

From http://www.reddit.com/r/CasualMath/comments/2jhwtm/is_fn_fn1_1fn1_f0_1_logarithmic_growth/

Question

What kind of growth does this function have? $f(n) = f(n-1) + \dfrac{1}{f(n-1)};~n\in\mathbb{N}$, where $f(0) = 1$

From http://www.reddit.com/r/CasualMath/comments/2jhwtm/is_fn_fn1_1fn1_f0_1_logarithmic_growth/

geerky42 · Answer

One answer said it seemingly approaches to $\sqrt{2x}$, so $O(\sqrt x)$

But how can we show that?

I don't understand other answer, though.

geerky42 · Answer

Do you have any idea? @SithsAndGiggles