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.