We　investigate　the　price　of　anarchy　(PoA)　in　nonatomic　congestion　games　when　the　total　demand　T　gets　very　large.　First　results　in　this　direction　have　recently　been　obtained　by　Colini-Baldeschi　et　al.　(2016,　2017,　2020)　for　routing　games　and　show　that　the　PoA　converges　to　one　when　the　growth　of　the　total　demand　T　satisfies　certain　regularity　conditions.　We　extend　their　results　by　developing　a　new　framework　for　the　limit　analysis　of　the　PoA　that　offers　strong　techniques　such　as　the　limit　of　games　and　applies　to　arbitrary　growth　patterns　of　T.　We　show　that　the　PoA　converges　to　one　in　the　limit　game　regardless　of　the　type　of　growth　of　T　for　a　large　class　of　cost　functions　that　contains　all　polynomials　and　all　regularly　varying　functions.　For　routing　games　with　Bureau　of　Public　Road　(BPR)　cost　functions,　we　show　in　addition　that　socially　optimal　strategy　profiles　converge　to　equilibria　in　the　limit　game　and　that　the　PoA　converges　to　one　at　a　power　law　with　exponent　beta,　where　beta　＞　0　is　the　degree　of　the　BPR　functions.　However,　the　precise　convergence　rate　depends　crucially　on　the　the　growth　of　T,　which　shows　that　a　conjecture　proposed　by　O＇Hare　et　al.　(2016)　need　not　hold.