Compositions of linear-fractional transformations

We study the asymptotic behavior of the compositions $(\mathbf S_n\circ…\circ\mathbf S_1)(z)$ and $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ of linear-fractional transformations $\mathbf S_n(z)$ ($n=1,2,…$) whose fixed points have limits. In particular, if $\mathbf S_n(z)=\alpha_n(\beta_n+z)^{-1}$, then the sequence of compositions $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ at the point $z=0$ coincides with the sequence of convergents of the formal continued fraction
$$ \frac{\alpha_1}{\beta_1+\dfrac{\alpha_2}{\beta_2+\dotsb}}. $$
The result obtained can be applied in the study of convergence of formal continued fractions.