In　this　letter,　we　analyze　the　downlink　performance　of　ultra-dense　networks　(UDNs)　considering　a　multi-piece　path　loss　model　with　line-of-sight　(LoS)　and　non-LoS　(NLoS)　transmissions,　a　general　multi-path　fading,　three　dimensional　transmission　distances　and　a　finite　user　equipment　(UE)　density.　A　constant　capacity　scaling　law　is　obtained,　that　is,　both　the　coverage　probability　and　the　area　spectral　efficiency　approach　a　constant　when　the　base　station　(BS)　density　grows　extremely　large.　However,　this　performance　limit　is　not　always　the　maximum　achievable　performance,　which　means　that　network　densification　alone　is　not　always　the　best　strategy.　Moreover,　the　performance　difference　of　various　multi-path　fading　models　is　more　conspicuous　with　the　growth　of　BS　density,　and　highly　depends　on　the　UE　density.