We consider an ergodic stochastic control problem for a class of one-dimensional Ito processes where the available control is an added bounded variation process. The corresponding in finite horizon discounted control problem was solved in [ A. Weerasinghe, SIAM J. Control Optim., 44 ( 2005), pp. 389-417]. Here, we show that as the discount factor approaches zero, the optimal strategies derived in [ A. Weerasinghe, SIAM J. Control Optim., 44 ( 2005), pp. 389-417] "converge" to an optimal strategy for the ergodic control problem. Under different assumptions, two types of optimal strategies were derived. Also, the Abelian limit relationships among the ergodic control problem, the in finite horizon discounted control problem, and the finite time horizon control problem are established here. A solution to a constrained optimization problem is obtained as an application.