We study geometric limits of convex-cocompact cyclic subgroups the rank 1 groups SO_0(1, k+1) and SU(1, k+1). construct examples sequences such G that converge algebraically whose limit strictly contains algebraic limit, thus generalizing example first described by Jorgensen for SO_0(1,3). also give necessary sufficient conditions a subgroup to arise as sequence subgroups. then discuss generali...