'asymptote' originates from Latin and Greek, specifically from Late Latin 'asymptota' and Greek 'asymptōtos', where the prefix 'a-' meant 'not' and 'symptōtos' (from 'sympiptein' / 'sumpiptein') related to 'falling together' or 'meeting'.

'asymptote' came into English via Late Latin 'asymptota' and Medieval/Modern French forms, ultimately from Greek 'asymptōtos' (ἀσύμπτωτος); the Greek term was adopted in scientific Latin and then borrowed into English.

Initially, it meant 'not falling together' (i.e., not meeting); over time it became the technical mathematical term for a line that a curve approaches but does not meet, and later was also used figuratively for an unattainable limit.