In this study, the existence and stability of dark solitons in the discrete nonlinear Schr\{o}dinger (DNLS) equation in the continuum limit (strong coupling between lattice sites) are analyzed. Our analysis shows that the DNLS admits only two types of solutions, namely the onsite and intersite solitons corresponding to the soliton`s centre being at a lattice site and between lattice sites, respectively. While numerical studies have been done to show the instability of both solitons, an analytical study of their instability has yet to be examined. We develop an exponential asymptotics/asymptotics beyond all orders method to analyze the instability of the intersite solitons. Our analytical prediction shows excellent agreement with the numerical results. We also extend our results to study the existence and stability of discrete bright solitons in the DNLS. Our application shows the versatility of this approach for analyzing multiscale problems in discrete nonlinear systems.