Algorithm for finding the limit of unkown function
This is some form of fixed point iteration (you are going $x_{n + 1} = f(x_n)$, and looking for $x^* = f(x^*)$). If you know that the convergence is linear (i.e., $e_{n + 1} \approx C e_n$, where $e_n$ is the $n$-th error), you can accelerate the convergence by Aitken's $\Delta^2$ method. Say you have $x_0$, $x_1$ and $x_2$. A (hopefully much better) approximation is given by:

$\begin{align*}
  x^+
    &amp;= x_2 - \frac{(\Delta x_1)^2}{\Delta^2 x_0} \\
    &amp;= x_2 - \frac{(x_2 - x_1)^2}{x_2 - 2 x_1 + x_0}
\end{align*}$

Start with $x_0 = x^+$ and compute new values for $x_1, x_2$. Repeat until satisfactory precision.