  We focus on a class of Weierstrass elliptic threefolds that allows the base
of the fibration to be a Fano surface or a numerically $K$-trivial surface. We
define the notion of limit tilt stability, and show that the Fourier-Mukai
transform of a slope stable torsion-free sheaf satisfying a vanishing condition
in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We
also show that the inverse Fourier-Mukai transform of a limit tilt semistable
object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to
modification in codimension 2.
