integrable lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of the macroscopic effective theory. these models have found broad applications in condensed matter physics and numerical analysis. in this talk, i will discuss our recent work on the continuum limit of some differential-difference equations. using the ablowitz--ladik system (al) as our prototypical example, we establish that solutions to this discrete model converge to solutions of the cubic nonlinear schr\"odinger equations (nls). notably, we consider merely $l^2$ initial data which combines both slowly varying and rapidly oscillating components, and demonstrate convergence to a decoupled system of nls. this surprising result highlights that a sole nls does not suffice to encapsulate the al evolution in such a low-regularity setting reminiscent of the thermal equilibrium state. i will also explain the framework of our proof and how it has been successfully extended to address more general lattice approximations to nls and mkdv.