Lime scale is a common problem in places with hard water. Over time, items in your home which come into contact with the water may develop a white, chalky layer of lime or become dull and unappealing in appearance. Lime is an alkaline substance, meaning that it's a base as opposed to an acid. Many mild acids will safely clean lime off of metals, such as copper. You have several options that you can try, all of which use materials that you probably already have in your home.
Cut a lemon in half, then rub some salt on the inside of one half. Rub the copper with the lemon and salt, then rinse with warm water. The mild acid is safe for your copper and will dissolve lime; it's also safer than using a more abrasive material, which could scratch the relatively soft copper. Don't be too aggressive with the salt scrub for just this reason. This is a good way to remove light deposits and to safely shine the copper at the same time.
Mix a paste of baking soda and vinegar, using about 3 parts baking soda to 1 part vinegar. Rub the paste onto the dull or chalky spots. It's safe to leave the paste on the lime for a little while to help dissolve it. Scrub and wipe away with a soft, damp cloth; again, don't use something abrasive. This method is well suited to thicker or difficult-to-remove lime deposits.
Soak the copper in vinegar to get rid of tough or stubborn lime deposits. Pickling vinegar is stronger that the regular kind, and you may find it better suited for dissolving tough deposits. If you can't soak the item due to size or if it's permanently attached to something else, soak a soft cloth or paper towel in vinegar and wrap it around the item. Rinse with warm water afterward.