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Between new moon and full moon, the moon's disc as viewed from the earth is bounded by a semicircular arc on its right and an arc on its left that moves to the right until at full moon it forms the other part of the circle.

Between full moon and new moon, the disc is bounded by a semicircular arc on its left and an arc on its right that moves until it coincides with the leftside arc, making the visible moon disappear in the sky.

Is there a word or short phrase that denotes the apparently mobile part of the lunar limb?

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It's called "the terminator line" or simply "the terminator"; where the illumination from the sun terminates.

There's a fuller discussion in EarthSky.org's What is the moon’s terminator line?, with ends:

Bottom line: The terminator line – on Earth’s moon and other planets and moons in space – is caused by sunlight falling on the surfaces of these worlds. It’s the line dividing night and day on the moon and other solar system objects.

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