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As value of $g$ is maximum on the surface of the earth and zero at the center. My question is what is the value of distance from surface of Earth where value of $g$ is zero? I tried to solve the problem by using Newton's law of the universal gravitation but it gives the infinity as answer,but according to my thought there must be a distance (limit) after that value of $g$ become zero.

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In a simple universe where the only thing that had gravity on you was the Earth, there is no such limit. As you seem to have seen from Newton's law of gravitation, there is no finite $R$ such that $R^{-2}$ is zero, and that's the only thing that changes. In real life, you can find places where other objects in space have equal but opposite gravitational force, and in these places the force cancels and there $g = 0$. See Lagrange points.

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