How far from the Lunar surface does gravity become 0?

Answers (1)

Gravity never becomes zero. What you are looking for is the distance where attraction of the Earth balances attraction of the moon. Here are some links that will lead you to that answer while explaining some other related things too:
www.google.com/search?q=moon%27s+gravity+relative+to+earth&ie=utf-8&oe=utf-8

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