"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.