This　paper　concerns　the　relaxation　time　limits　for　the　one-dimensional　steady　hydrodynamic　model　of　semiconductors　in　the　form　of　Euler-Poisson　equations　with　sonic　or　nonsonic　boundary.　The　sonic　boundary　is　the　critical　and　difficult　case　because　of　the　degeneracy　at　the　boundary　and　the　formation　of　the　boundary　layers.　In　order　to　avoid　the　degeneracy　of　the　second　order　derivatives,　we　technically　introduce　an　invertible　transform　to　the　working　equation.　This　guarantees　that　the　remaining　one　order　degeneracy　becomes　a　good　term　since　the　transform　used　here　is　strictly　increasing.　Then　we　efficiently　overcome　the　degenerate　effect.　When　the　relaxation　time　tau　-＞　+infinity,　we　first　show　the　strong　convergence　of　the　approximate　solutions　to　their　asymptotic　profiles　in　L-infinity　norm　with　the　order　O(tau(-1/2)).　When　tau　-＞　0(+,)　the　boundary　layer　appears　because　the　boundary　data　are　not　equal　to　each　other,　and　we　further　derive　the　uniform　error　estimates　of　the　approximate　solutions　to　their　background　profiles　in　L-infinity　norm　with　the　order　O(tau(1/2))　or　O(tau)　according　to　the　different　cases　of　boundary　data.　Unlike　the　methods　adopted　in　the　previous　studies,　we　propose　some　altogether　new　techniques　of　the　asymptotic　limit　analysis　to　successfully　describe　the　width　of　the　boundary　layer,　which　is　almost　the　order　O(tau)　provided　0　＜　tau　＜＜　1.　These　original　approaches　develop　and　improve　the　existing　studies.　Finally,　some　numerical　simulations　are　carried　out,　which　confirm　our　theoretical　study,　in　particular,　the　appearance　of　boundary　layers.