We study the power of four popular unit root tests in the presence of a local-to-finite variance DGP. We characterize the asymptotic distribution of these tests under a sequence of local alternatives, considering both stationary and explosive ones . We supplement the theoretical analysis with a small simulation study to assess the finite sample power of the tests. Our results suggest that the finite sample power is affected by the $\alpha$-stable component for low values of $\alpha$ and that, in the presence of this component, the DW test has the highest power under stationary alternatives. We also document a strange behavior of the $DW$ test which, under the explosive alternative, suddenly falls from 1 to zero for very small changes in the autoregressive parameter suggesting a discontinuity in the power function of the $DW$ test