Here, the singularly perturbed system is considered: \[ \dot x(t) =F(t,x_t,y_{t, \varepsilon}),\;x(t)\in \mathbb{R}^n,\quad \varepsilon\dot y(t)=g (t,x_t,y_{t, \varepsilon}),\;y(t)\in \mathbb{R}^m, \] with \(x_t(\theta) =x(t+\theta)\), \(y_{t,\varepsilon} (\theta)= y(t+\varepsilon \theta)\), \(\theta\in [-\tau,0]\). Under certain conditions, the following result is obtained: there exists a sequence \(\varepsilon_i \to\infty\) such that \(x_{\varepsilon_i}\) converges uniformly, and \(y_{\varepsilon_i}\) converges narrowly to a Young measure. As an application, some concrete examples are examined.