Before spending money on lime I'd spend a few £s on a PH test - lime will only work to fix PH, we have bad buttercups in one field but the PH is 7 so lime wouldn't make any difference, the species of buttercups we have clearly don't mind the PH!! .
It really depends on the ph of your soil. To be honest I don't think liming gets rid of buttercups. I know my soil is fine but I still have buttercups.