How much range would a 3 inch 20ga slug be lethal for?
I think a slug will be lethal out way past it's usable range.
I believe the limiting factor would actually be the accuracy of the gun and the skill of the shooter, not the 'lethality' of the slug. Slugs will start dropping like rocks because of their fairly poor ballistic coefficients.
Also, depending on how accurate the load/gun is, past a certain range, your groups will be too large to effectively and humanely harvest an animal. If you can only manage 3'' groups @ 100 yards, 200 yards your groups will be at least twice as large.
I'd say a slug could be lethal past 400 yards pretty easily, if it hit the right spot. The trouble would be getting it into that specific spot.
Most shotguns and the people operating them, will not be able to shoot past 100 yards.
Just to put it into perspective. A 250 grain SST 20ga slug will cross the sound barrier @ roughly 250 yards. That means the slug still has more mass and velocity (and therefore energy) @ 250 yards, than a .45ACP +P+ load has at the muzzle..
560 yards the slug is traveling about 830FPS
That's the speed of a standard .45ACP military load, at the muzzle, still with 20 more grains of bullet mass...