This　work　is　concerned　with　the　two-fluid　Euler-Maxwell　equations　for　plasmas　with　small　parameters.　We　study,　by　means　of　asymptotic　expansions,　the　zero-relaxation　limit,　the　non-　relativistic　limit　and　the　combined　non-　relativistic　and　quasi-neutral　limit.　For　each　limit　with　well-prepared　initial　data,　we　show　the　existence　and　uniqueness　of　an　asymptotic　expansion　up　to　any　order.　For　general　data,　an　asymptotic　expansion　up　to　order　1　of　the　non-　relativistic　limit　is　constructed　by　taking　into　account　the　initial　layers.　Finally,　we　discuss　the　justification　of　the　limits.