adjective Mathematics.
of or pertaining to an asymptote.
(of a function) approaching a given value as an expression containing a variable tends to infinity.
(of two functions) so defined that their ratio approaches unity as the independent variable approaches a limit or infinity.
(of a formula) becoming increasingly exact as a variable approaches a limit, usually infinity.
coming into consideration as a variable approaches a limit, usually infinity: asymptotic property; asymptotic behavior.
Also, asymptotical.

1665–75; asymptote + -ic

asymptotically, adverb Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
asymptotic or asymptotical (ˌæsɪmˈtɒtɪk)
1.  of or referring to an asymptote
2.  (of a function, series, formula, etc) approaching a given value or condition, as a variable or an expression containing a variable approaches a limit, usually infinity
asymptotical or asymptotical
asymptotically or asymptotical

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Example sentences
So, it seems to me that without altering the brain machinery, our social
  progress will asymptotically approach a limit.
When transaction costs goes closer and closer to zero, trading volume would
  asymptotically go closer and closer to infinity.
So the pressure increases asymptotically with depth up to a threshold.
Asymptotically, therefore, the conditional distribution of given satisfies.
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