To a system of differential algebraic equations: \[ \text{(DAE)}\quad y'(t)=f(t,y(t),z(t),0),\quad g(t,y(t),z(t),0)=0, \] a system of singularly perturbed ordinary differential equations: \[ \text{(ODE)}\quad y_ \varepsilon'(t)=f(t,y_ \varepsilon(t),z_ \varepsilon(t),\varepsilon), \varepsilon z_ \varepsilon'(t)=g(t,y_ \varepsilon(t),z_ \varepsilon(t),\varepsilon) \] is assigned. Under certain conditions, and in certain sense, the solutions \(y_ \varepsilon\) and \(z_ \varepsilon\) of the singularly perturbed system (ODE) converge to \(y\) and \(z\) respectively, when \(\varepsilon\to 0\). However, in general the system (ODE) is stiff: its stiffness increases with \(\varepsilon\) diminishing. In order to resolve approximately the system (DAE) one may try to apply certain numerical methods, suitable to stiff systems (for example \(A\)- stable methods) to the system (ODE) for \(\varepsilon\) sufficiently small. The obtained equations will contain two small parameters: \(\varepsilon\) and the step of integration \(h\). The main result of this paper consists in giving sufficient conditions, for implicit Runge-Kutta methods, for the convergence of the solution of the discretized, singularly perturbed system (ODE) to the solution of (DAE). However the main goal is that the estimates for the error bounds are uniform in the sense, that one may pass to the limit with \(\varepsilon\to 0\), uniformly with respect to \(h\) and vice versa. The author obtains this result because the method of so-called \(B\)- convergence is applied in the proof. Proofs are given in extenso, with some reference to the literature. Certain known methods are discussed. Results of this work are compared with results in the same field, obtained recently by other authors.