I've never seen any firm data on how deep to disc it in. You don't want it too deep. If your lime goes deeper than the plant roots will go, it's of no benefit. I'd say the top few inches of soil is where you need the lime. That's where the plant roots will be. You don't necessarily have to disc it in. I would certainly disc it in if the plot had a lot of slope to it. You don't want all your lime washing away. If the plot is flat and the lime is not likely to wash away, I'd just scratch the surface to allow some lime to get down a few inches deep.