It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{2}}{2x + \frac{5^{2}}{2x + \frac{7^{2}}{2x + \cdots}}}}$$

Are there any other $x$ that $C(x)$ can be approximated through AGM quickly? In general is there a comprehensive study between Infinite Continued fractions and their calculations through AGM?

Why does AGM work at $x=1$?

Is there any connection to elliptic integrals?