Exhausted from effort, especially from talking, arguing, or trying to convince someone of something without success. Can also literally refer to a bluish complexion from lack of oxygen.
This phrase comes from the medical observation that people turn blue (cyanotic) when they're not getting enough oxygen, whether from holding their breath, choking, or extreme exertion. The metaphorical use developed to describe futile efforts that leave someone as exhausted as if they'd been deprived of air.
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