et X be a smooth projective threefold where the generalized Bogomolov-Gieseker inequality holds. In this paper we examine the conditions that an object E with ch0(E) = 0 has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to equivalent to sheaf Giesker- Simpson (semi)stability or a dual of it, depending on the curve considered.