What is the limit set of a subgroup $G$ of Möbius transformations?

Suppose that $G$ is complicated and has no finite orbit in $\mathbb{C} \cup\{\infty\}$. Prove that the limit set of $G$ is infinite. Can the limit set be countable?

State Jørgensen's inequality, and deduce that not every two-generator subgroup $G=\langle A, B\rangle$ of Möbius transformations is discrete. Briefly describe two examples of discrete two-generator subgroups, one for which the limit set is connected and one for which it is disconnected.