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In this paper we establish computable conditions for the stability and instability of limit cycles in nonlinear feedback systems. In the proof of the present results, we make use of several novel transformations, of averaging, and of a result on integral manifolds and we assume that we can establish the existence of limit cycles by means of the describing function method. Our results, which in part justify the popular quasistatic stability analysis of limit cycles (Loeb's criterion), are significantly different from existing results dealing with the stability analysis of limit cycles. We demonstrate the applicability of our results by means of specific examples.