On a cutting board, cut the tops off the pineapples and discard. Cut a thin slice off the bottom of the pineapple so it will stand steadily on the cutting board. With a large knife, cut off the rind from the pineapples with as little pineapple attached as possible. Cut the pineapple in quarters lengthwise, then into 1-inch cubes.
Pass the pineapple and melon chunks through a juicer into a large measuring cup. Pass the juice through a fine strainer to remove any froth. Stir in the lime juice. Chill thoroughly before serving. The juice will keep 1 or 2 days in the refrigerator.
Tip: Most supermarkets now sell pineapple and melon that have been peeled and cut into cubes.