What does asymptotically mean?
Asymptotically is a formal adverb describing behavior relative to an asymptote — a line or value that a curve or quantity approaches ever more closely without ever actually reaching it. In mathematics and computer science it is indispensable for characterizing limits: a sequence may grow asymptotically toward a bound, an error estimate may shrink asymptotically to zero, and one algorithm may be asymptotically faster than another, meaning the advantage emerges as input sizes grow very large. The word carries a precise, technical connotation of convergence without arrival, and this image has given it a modest figurative life beyond mathematics, where writers use it to describe efforts that come tantalizingly close to a goal yet fall short by definition. Rooted in the Greek asymptōtos ('not falling together'), it belongs chiefly to academic and scientific registers and rarely appears in everyday conversation.
adverbIn a manner that relates to an asymptote or to the limiting behavior of a function or process as a variable approaches infinity or some limiting value.
- With reference to the limiting behavior of a mathematical function, sequence, or process as a variable tends toward infinity or another limit.
- By extension, in general usage: tending ever closer to something without ever quite reaching it.
"The algorithm runs asymptotically faster than its predecessor, though real-world gains only appear at very large input sizes."
"The error term shrinks asymptotically to zero as the sample size increases."
"Their two estimates converge asymptotically, never quite agreeing."
Two algorithms can both 'work' — but mathematicians will tell you one is asymptotically faster, and at scale, that difference decides which survives.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of asymptotically
Asymptotically derives from asymptotic, the adjective form of asymptote, which entered English from Greek asymptōtos, meaning 'not falling together' or 'not meeting' — from a- ('not') plus syn ('together') and piptein ('to fall'). The geometric term was adopted into Latin and then English in the 17th century as mathematicians studied curves such as the hyperbola that approach a straight line indefinitely without touching it. Cognates sharing the same Greek root include symptom and symptomatic, which preserve the sense of 'falling' or 'occurring' in the context of disease. The adverb form arose later, as the adjective's technical use expanded into modern analysis and complexity theory.
Related word forms
How asymptotically is actually used
Primarily a technical term used in mathematics, computer science, physics, and statistics; outside these fields it appears mostly in figurative or metaphorical writing about things approaching a limit. Register is formal or academic. It modifies verbs, adjectives, and other adverbs (e.g., 'asymptotically approach,' 'asymptotically equivalent').
Easily confused with asymptotically
Asymmetrically means in a way lacking symmetry between parts, while asymptotically concerns approaching a limit or asymptote.