This　paper　is　concerned　with　the　relaxation-time　limits　to　a　multidimensional　radial　steady　hydrodynamic　model　of　semiconductors　in　the　form　of　Euler-Poisson　equations　with　sonic　or　nonsonic　boundary　as　the　relaxation　time　\tau　\rightarrow　\infty　and　\tau　\rightarrow　0+,　respectively,　where　the　sonic　boundary　is　the　critical　and　difficult　case,　because　of　the　degeneracy　at　the　boundary　and　the　formation　of　boundary　layers.　For　the　case　of　\tau　\rightarrow　\infty　,　after　showing　the　boundedness　of　the　density　by　using　the　divergence　form,　we　prove　the　convergence　of　the　solutions　to　their　nontrivial　asymptotic　states　with　the　convergence　order　O(\tau　-　2　)　in　the　Lo　degrees-sense.　In　order　to　overcome　the　degeneracy　caused　by　the　critical　sonic　boundary,　we　introduce　an　inverse　transform　as　a　technical　tool　to　remove　the　secondorder　degeneracy,　and　observe　the　advantage　of　a　first-order　degeneracy　due　to　the　monotonicity　of　this　transformation.　Moreover,　when　\tau　\rightarrow　0+　with　different　boundary　values,　where　the　boundary　layers　appear,　we　show　the　strong　convergence　order　O(\tau　)　or　O(\tau　1-\varepsilon　)　for　different　boundary　cases.　In　order　to　overcome　the　difficulty　caused　by　the　boundary　layer,　we　propose　a　new　technique　in　asymptotic　limit　analysis　and　identify　the　width　of　the　boundary　layers　as　O(\tau　).　These　new　proposed　methods　develop　and　improve　upon　the　existing　studies.　Finally,　a　series　of　numerical　simulations　are　conducted,　which　corroborate　our　theoretical　analysis,　particularly　regarding　the　formation　of　boundary　layers.