Phrase at a loss "confused, uncertain" (1590s) is a phrase from hunting, in reference to hounds losing the scent. To cut (one's) losses is from 1885, originally in finance. The retailer's loss-leader "advertised product sold at cost or below" (to entice customers in to buy other things as well) is from 1922.
Old English losian "be lost, perish," from los "destruction, loss," from Proto-Germanic *lausa- (source also of Old Norse los "the breaking up of an army;" Old English forleosan "to lose, destroy," Old Frisian forliasa, Old Saxon farliosan, Middle Dutch verliesen, Old High German firliosan, German verlieren, as well as English -less, loss, loose). The Germanic word is from PIE *leus-, an extended form of root *leu- "to loosen, divide, cut apart."
The verb also is merged with, or has taken the (weaker) sense of, the related Middle English leese "be deprived of, lose" (Old English leosan, a class II strong verb whose past participle loren survives in forlorn and love-lorn), from Proto-Germanic *leusanan (source also of Old High German virliosan, German verlieren, Old Frisian urliasa, Gothic fraliusan "to lose").
Hence lose in the transitive senses "part with accidentally, be deprived of, miss the possession or knowledge of" (money, blood, sleep, hair, etc.), c. 1200; "fail to keep, lose track of" (mid-13c.). Meaning "fail to preserve or maintain" is from mid-15c. Meaning "fail to gain or win" (something) is from c.1300; intransitive meaning "fail to win" (a game, contest, lawsuit, etc.) is from late 14c. Meaning "to cause (someone) to lose his way" is from 1640s; meaning "cease to have, be rid of" (something unwanted) is from 1660s.
To lose heart "become discouraged" is from 1744; to lose (one's) heart "fall in love" is from 1630s. To lose (one's) mind "become insane" is attested from c. 1500. To lose out "fail" is 1858, American English. To lose it "become distraught, break down and lose control of oneself" is by 1990s; the it probably being one's self-control or grip on reality. Related: Lost; losing.
late 13c., "death, act of expiring, loss of life," verbal noun from die (v.).
"despondent condition, a sinking or dejection of spirit from loss of hope or courage in affliction or difficulty," 1670s, from Latin despondentem (nominative despondens), present participle of despondere "to give up, lose, lose heart, resign," also "to promise in marriage" (especially in phrase animam despondere, literally "give up one's soul"), etymologically "to promise to give something away," from de "away" (see de-) + spondere "to promise" (see sponsor (n.)).
Despondency is a loss of hope sufficient to produce a loss of courage and a disposition to relax or relinquish effort, the despondent person tending to sink into spiritless inaction. Despair means a total loss of hope; despondency does not. [Century Dictionary, 1897]
