Cognitive Radio Networks have emerged in the last decades as a solution of two problems: spectrum underutilization and spectrum scarcity. In this work, we propose a dynamic spectrum sharing mechanism, where primary users have strict priority over secondary ones in order to improve the mean spectrum utilization with the objective of providing to secondary users a satisfactory grade of service with a small interruption probability. We study a stochastic model for Cognitive Radio Networks with fluid limits techniques. Our main findings consist in a Gaussian limit theorem in the sub-critical case, and a non-Gaussian limit theorem, under a different scaling scheme, in the critical case. These results provide us practical QoS criteria for sharing policies. We support our analysis with representative simulated examples in both scenarios.