This study investigates the convergence of solutions to boundary-value problems for dynamic equations defined on a family of time scales toward solutions of corresponding boundary-value problems for differential equations on the real interval. We establish conditions under which a solution exists for the boundary-value problem on the real axis, provided that such a solution exists for a family of time scales characterized by a sufficiently small graininess function. We demonstrate the convergence of solutions on time scales to those on the real interval, providing estimates of the convergence rate. Our findings contribute to the understanding of the interrelation of dynamic equations on discrete and continuous domains.