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why doesn't gravity speed up light?

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We know that gravity speeds up a body, like a meteor which enters the earth gets constantly speeded up by earth's gravity.
And from relativity we know that light bends near a massive body because newton's law of gravitation is just an approximation and actually gravity depends on energy and momentum.
So my question is: If a ray of light is aimed exactly at the centre of a body, then will it get accelerated like a meteor? And if does get accelerated, then won't it surpass the universal speed limit of 3,00,000 km/s (approx.)?

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