This paper describes a method to detect limit cycles for optimal control problems in the plain. The procedure includes two steps. First, the solution paths are analytically studied for large discount rates. Second, we demonstrate by means of computer simulations how the dynamics found can be traced back to small discount rates. We apply this method to two specific examples from resource management : a taxation problem and an exploited system of predator-prey interaction which show that the limit cycles may grow as the discount rates decrease. The principle that small discount rates are more conservative than large ones is therefore questionable. The relation of our results to theorems in optimal growth theory is also discussed.