Frequency Response?

If a speaker is rated down to a given frequency response (say 40 hz), wouldn't that suggest that 40 hz is about as low as the given driver is capable of reproducing, not accounting for roll off? When taking roll off into concideration, will the driver continue to attempt to reproduce frequencies infinately below the given 40 hz, or is there a point that it is just incapible so it doesn't? What I am asking is, Will a driver designed and constructed to play down to 40 hz even attempt to reproduce a 20 hz signal? I want answers!