These relations are extremely useful, and even allow us to extract exponents and . But only if we are at ``large enough'' L. If is very large (but not infinite), may get incorrect ``pseudo-critical'' exponents.
This is often the case at a ``weakly'' first order transition (very small latent heat). So it is often very difficult to distinguish between a first and second order phase transition, or even to say for sure if there is a phase transition at all --- is , or just ? May need extremely large system sizes to get correct results.
In general, don't know the form of the finite size scaling, so try to fit data to a scaling function, or go to large enough systems so that results constant or can do a simple linear extrapolation.