The moon is slowly moving away from the earth. what will eventually happen to the gravitational force between the earth and the moon? why?

Well, recall Newton's Law of Gravity.  It states the gravitational force between two objects is directly proportional to the respective masses and inversely proportional to the square of the distances between their respective centres of mass.  Thus, as the distance between the object's centres of mass increases, so does the square, therefore the gravitational attraction decreases.