brink of death; one's last moments
Refers to the moment just before death. Often used in Buddhist or literary contexts.
He chanted a Buddhist prayer on his deathbed.
knowing when to give up; grace in defeat
Almost exclusively used in the set phrase 往生際が悪い, meaning 'not knowing when to give up' or 'being a poor loser'. The standalone noun is rare.
See also: 往生際が悪い
He was a poor loser and kept making excuses until the very end.
A graceless attitude in defeat is unsightly.