  Inspired by questions of convergence in continued fraction theory, Erd\H{o}s,
Piranian and Thron studied the possible sets of divergence for arbitrary
sequences of M\"obius maps acting on the Riemann sphere, $S^2$. By identifying
$S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we show
that these sets of divergence are precisely the sets that arise as conical
limit sets of subsets of $H^3$. Using hyperbolic geometry, we give simple
geometric proofs of the theorems of Erd\H{o}s, Piranian and Thron that
generalise to arbitrary dimensions. New results are also obtained about the
class of conical limit sets, for example, that it is closed under locally
quasisymmetric homeomorphisms. Applications are given to continued fractions.
