A class of second order nonlinear differential equations with delay depengingon the unknown function of the from(r(t)ψ(x(t))x' (t))' + f (t, x(t), x (△ (t, x(t)))) = 0in the case where ∫∞0 ds/r(s) ＜∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.