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This question already has an answer here:

If you have the minhag to eat in the Sukkah on Shmini Atzeres, why is there no minhag to take the lulav (without a bracha) on Shmini Atzeres?

merged by Double AA Oct 7 '14 at 5:53

This question was merged with Why don't those outside of Israel shake Lulav on Shmini Atzeres? because it is an exact duplicate of that question.