Middle English, from Old English lifian (Anglian), libban (West Saxon) "to be, be alive, have life; continue in life; to experience," also "to supply oneself with food, procure a means of subsistence; pass life in a specified fashion," from Proto-Germanic *libejanan (source also of Old Norse lifa "to be left; to live; to live on," of fire, "to burn;" Old Frisian libba, German leben, Gothic liban "to live"), from PIE root *leip- "to stick, adhere," forming words meaning "to remain, continue."
Meaning "to make a residence, dwell" is from c. 1200. Meaning "express in one's life" (live a lie) is from 1540s. Intensified sense "have life abundantly, make full use of life's opportunities" is from c. 1600. Related: Lived; living.
To live it up "live gaily and extravagantly" is from 1903. To live up to "act in accordance with, not live below the standard of" is 1690s, from earlier live up "live on a high (moral or mental) level" (1680s). To live (something) down "cause (something disreputable) to be forgotten by subsequent blameless course, live so as to disprove" is from 1842. To live with "cohabit as husband and wife" is attested from 1749; sense of "to put up with" is attested from 1937. Expression live and learn is attested from c. 1620.
According to the Dutch Prouerbe ... Leuen ende laetan leuen, To liue and to let others liue. [Gerard de Malynes, 1622]