In the linear fractional transformation $T(z)=\frac{az+b}{cz+d}$, if we define $T(-\frac{d}{c})=\infty$, then what is $T(\infty)$?
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There is no need to "define" these things... since, in fact, $G=GL\_2(\mathbb C)$ acts on complex projective one-space, a.k.a. the Riemann sphere, in homogeneous coordinates: coordinates are $\pmatrix{\omega\_1\cr \omega\_2}$ mod scalars $\mathbb C^\times$. $\mathbb C$ imbeds by $z\to \pmatrix{z \cr 1}$. Then the action of $G$ is just by matrix multiplication of vectors. Thus, the point of the projective space missing from $\mathbb C$ is $\pmatrix{1\cr 0}$. It is mapped to $\pmatrix{a \cr c}$ by $\pmatrix{a & b \cr c & d}$, etc.
$$\lim\_{z\to\infty}T(z)=\lim\_{z\to 0}T(1/z)=\lim\_{z\to 0}\frac{a/z+b}{c/z+d}=\lim\_{z\to 0}\frac{a+bz}{c+dz}=a/c.$$