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Limit screws and barrel adjuster.
I am confused when to use the L and H adjusters or use the barrel
tensioner/adjuster when adjusting my gears on my road bike.
Setting up a properly functioning Read Derailleur (RD) has a number of steps that need to be observed to end up with a successful outcome. The L and H screws and the barrel adjuster are really only 3 parts of a multi step process. This is a hard one to answer without actually showing you how to do it. In a nutshell though the L (low) and H (high) screws set the maximum lateral travel for the RD, and the Barrel Adjuster allows for micro adjustments of the shifter cable tension. The L and H screws should be set first and then the cable can be attached to the RD, where the barrel can be adjusted to achieve proper cable tension.