To completely remove or get rid of something. It can also mean to knock someone out of a competition so they can no longer continue.
From Latin 'eliminare' meaning 'to turn out of doors, banish,' from 'e-' (out) + 'limen' (threshold, doorway). The original sense was physically sending something or someone out of the doorway. Over time, it broadened to mean removing something in any way.
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