We analyze the asymptotic performance of a generalized multiuser diversity scheme for an interference-limited secondary multiuser network of underlay cognitive radio systems. Assuming a large number of secondary users and that the noise at each secondary user's receiver is negligible compared to the interference from the primary transmitter, the secondary transmitter transmits information to the $k$-th best secondary user, namely, the one with the $k$-th highest signal-to-interference ratio (SIR). We use extreme value theory to show that the $k$-th highest SIR converges uniformly in distribution to an inverse gamma random variable for a fixed $k$ and large number of secondary users. We use this result to derive asymptotic expressions for the average throughput, effective throughput, average bit error rate and outage probability of the $k$-th best secondary user under continuous power adaptation at the secondary transmitter, which ensures satisfaction of the instantaneous interference constraint at the primary receiver caused by the secondary transmitter. Numerical simulations show that our derived asymptotic expressions are accurate for different values of system parameters.