"part or member," Old English lim "limb of the body; any part of an animal body, distinct from the head and trunk;" main branch of a tree," from Proto-Germanic *limu- (source also of Old Norse limr "limb," lim "small branch of a tree"), a variant of *lithu- (source of Old English liþ, Old Frisian lith, Old Norse liðr, Gothic liþus "a limb;" and with prefix ga-, source of German Glied "limb, member").

The unetymological -b began to appear late 1500s for no etymological reason (perhaps by influence of limb (n.2)). The Old English plural was often limu; limen and other plural forms in -n lasted into Middle English. Since c. 1400 especially of a leg; in Victorian English this usage was somewhat euphemistic, "out of affected or prudish unwillingness to use the word leg" [Century Dictionary]. However in Old and Middle English, and until lately in dialects, it could mean "any visible body part":

The lymmes of generacion were shewed manyfestly. [Caxton, "The subtyl historyes and fables of Esope, Auyan, Alfonce, and Poge," 1484]

Hence, limb-lifter "fornicator" (1570s). Limb of the law was 18c. derisive slang for a lawyer or police officer. To go out on a limb in figurative sense "enter a risky situation" is from 1897. Alliterative life and limb in reference to the body inclusively is from c. 1200. Obsolete limb-meal (adv.) "limb-from-limb, piecemeal" is from late Old English lim-mælum.

"flaccid, drooping, lacking stiffness or firmness," 1706, of obscure origin, apparently from the first element in Old English lemphealt (see limp (v.)). Related: Limply; limpness. A limp wrist as indicative of male effeminate homosexuality is from 1960.

"thigh of a hog used for food" (especially salted and cured or smoke-dried), 1630s, extended from earlier sense of "part of the human leg behind the knee; hock of a quadruped," from Old English hamm "hollow or bend of the knee," from Proto-Germanic *hamma- (source also of Old Norse höm, Middle Low German, Middle Dutch hamme, Old High German hamma). This is from PIE *knam- "shin bone" (source also of Greek knēmē "calf of the leg, shank, tibia; spoke of a wheel," Old Irish cnaim "bone").

Ham-fisted (adj.) in reference to hard-hitting characters is from 1905; ham-handed "coarse, clumsy" is by 1896. With hammen ifalden "with folded hams" was a Middle English way of saying "kneeling."

dance in which the dancer bends backward and passes under a bar, 1956, of West Indian origin, probably an alteration of limber (adj.).

1760, "to free (a gun) from its limber," usually for the purpose of bringing it into action, from un- (2) "opposite of" + limber "attach a gun to its limber" (see limber (n.)). Figurative sense is attested from 1864. Related: Unlimbered; unlimbering.