How does $SL_2(\mathbb{R})$ act on the $\infty$ element?

Question

I am reading chapter 7 of J.P. Serre's Course in Arithmetic, and he states the following fact in the very first section:

"We make $SL_2(\mathbb{R})$ act on $\tilde{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ in the following way: if $g=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\in SL_2(\mathbb{R})$ and $z\in \tilde{\mathbb{C}}$, then we put $gz=\frac{az+b}{cz+d}$. How does this formula make any sense if $z=\infty$? How am I to interpret $gz?$

Answer 1

In that case it is the limit of $\frac{az+b}{cz+d}$ when $z\to\infty$.

Answer 2

I tell my students that "infinity arithmetic is allowed with fractional linear transformations". So $$\frac{a \infty + b}{c \infty+ d} = \frac{a + \frac{b}{\infty}}{c + \frac{d}{\infty}} = \frac{a+0}{c+0} = \frac{a}{c} $$ You can, of course, justify this formally using limits as in the answer of @Mark.