The reconfigurable intelligent surface (RIS) has drawn considerable attention for its ability to enhance the performance of not only the wireless communication but also the indoor localization with low-cost. This paper investigates the performance limits of the RIS-based near-field localization in the asynchronous scenario, and analyzes the impact of each part of the cascaded channel on the localization performance. The Fisher information matrix (FIM) and the position error bound (PEB) are derived. Besides, we also derive the equivalent Fisher information (EFI) for the position-related intermediate parameters. Enabled by the derived EFI, we verify that both the ranging and bearing information of the user can be obtained when the near-field model is considered for the RIS-User equipment (UE) part of the channel, while only the direction of the UE can be inferred in the far-field scenario. This result is well known in the scenario that the curvature of arrival (COA) is directly sensed by the traditional active large-scale array, and we prove that it still holds when the COA is sensed passively by the large RIS. For the base station (BS)-RIS part of the channel, we reveal that this part of the channel determines the type of the gain provided by the BS antenna array. Besides, in the single-carrier, single snapshot case, it requires both the BS-RIS and the RIS-UE part of the channel works in the near-field scenario to localize the UE. We also show that the well-known focusing control scheme for RIS, which maximizes the received SNR, is not always a good choice and may degrade the localization performance in the asynchronous scenario. The simulation results validate the analytic work. The impact of the focusing control scheme on the PEB performances under synchronous and asynchronous conditions is also investigated.