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On a Weierstra{\ss} elliptic surface $X$, we define a `limit' of Bridgelandstability conditions, denoted $Z^l$-stability, by varying the polarisation inthe definition of Bridgeland stability along a curve in the ample cone of $X$.We show that a slope stable torsion-free sheaf of positive (twisted) degree ora slope stable locally free sheaf is taken by a Fourier-Mukai transform on$D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzerofiber degree can be modified so that its inverse Fourier-Mukai transform is aslope semistable torsion-free sheaf.

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic surfaces