Consider a time-inhomogeneous branching random walk,generated by the point process Ln which composed by two independent parts:'branching'offspring Xn with the mean 1+B(1+n)-β for β∈(0,1)and'displacement'ζn with a drift A(1+n)-2α for α∈(0,1/2),where the'branching'process is supercritical for B＞0 but'asymptotically critical'and the drift of the'displacement'ζn is strictly positive or negative for |A|＞0 but'asymptotically'goes to zero as time goes to infinity.We find that the limit behavior of the minimal(or maximal)position of the branching random walk is sensitive to the'asymptotical'parameter β and α.