In my opinion, for your case you would recall the meaning of that formula which is Hooke's law:
F/A=E*(l-l0)/l0
When F->0 you get l->l0 which is simply "No force applied to, no elongation as effect". As math only, l->l0 means also F->0 but this is not cause-effect.
I cannot see any singularity in this physics.

If you try to investigate l0=0 you are losing the physics behind. Because you try to find what happened with a cylindrical body with zero initial length when apply an axial force; physically the force cannot elongate such cylinder because the cylinder does not exist there. As the cylinder has any length- even one incredible small for practice- physics works correctly. And in your query, when say l0 tends to zero (as limit) does not means l0=0 but l0 with a value near zero and physics works with, as above written.