Let $f(a,x)=\sqrt{a(a-1)+x}$, $f^{(n)}(a,x)$ denotes the $n^{th}$ iteration of $x$, where \begin{align}f^{(1)}(a,x)=f(a,x),f^{(n)}(a,x)=f^{(n-1)}(a,f(a,x))\end{align} Find \begin{align}\lim_{n\to\infty}(2a)^{\frac{n}{2}}\sqrt{a-f^{(n)}(a,0)}\end{align}
This is a problem from a discussion in a forum, someone say that when $a=2$, the limit is equal to $\pi$, but I can't find the pattern behind this structure, Thanks for your attention.
PS: Expanding it shows \begin{align}\lim_{n\to\infty}(2a)^{\frac{n}{2}}\sqrt{a-\sqrt{a(a-1)+\sqrt{a(a-1)+\cdots}}}\end{align} where there are $n$ square root sign behind the minus sign, it is a $\infty \cdot 0$ problem.