Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A straight line which a curve approaches arbitrarily closely, as they go to infinity. The limit of the curve, its tangent "at infinity".
 n. Anything which comes near to but never meets something else.
 v. To approach, but never quite touch, a straight line, as something goes to infinity.
from the GNU version of the Collaborative International Dictionary of English
 n. A line which approaches nearer to some curve than assignable distance, but, though infinitely extended, would never meet it. Asymptotes may be straight lines or curves. A rectilinear asymptote may be conceived as a tangent to the curve at an infinite distance.
from The Century Dictionary and Cyclopedia
 In mathematics, approaching indefinitely close, as a line to a curve, but never meeting. See II.
 n. A straight line whose distance from a curve is less than any assignable quantity, but which does not meet the curve at any finite distance from the origin.
 n. In zoology, either of two straight lines drawn from the apex to the aperture of the conical part of the phragmocone in the belemnites or fossil dibranchiate cephalopods.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a straight line that is the limiting value of a curve; can be considered as tangent at infinity
Etymologies
Examples

In that case, if you plot the number of record highs or lows on a graph, you'd get a line that falls gradually as time progresses toward an asymptote, which is mathematically expressed as 1/n, where n is the number of years in your data set.

Man is the asymptote of what he predicates God to be.

"What part of an inverse tangent function approaching an asymptote don't you understand?"
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For his opening move — in which "Oh" would have been a feasible if less canonic alternative (fully licensed by the dictionary) — is a line that negotiates in process between the vocal base line of expressive oralilty, on the near hand, and, at expression's farthest reach, the vocative asymptote of natural communion with inanimate energy.

Marked by the thickened release of "good" from "growing," what we find inscribed from within narrative time is both a phrase for cumulative social improvement and an asymptote of its visionary teleology as well, Tennyson secularized: the immediate "growing betterment" (participial adjective plus noun) as well as, hard on its heels, the "growing [ultimately] good"

Buchannan, at least, I think he can't displace; the man was practically an asymptote.

Then I walk around downtown Cleveland, using the View option to study a random timeline as far up the 300 year Masterson asymptote as I can get (usually at least a quarter millennium).
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Little did they know that they were just a literary device for explaining the asymptote opposite to Ralph and Piggy, the social cage where expected behavior overwhelms the individual totally.
All New York's a stage, and all the Mad Men and Women merely players

Like a curve forever approaching its asymptote but never touching it, I could potentially add to the list forever.

Obama's curve hits the vertical asymptote well before the Ohio and Texas primaries.
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jmjarmstrong commented on the word asymptote
JM draws a line at asymptote after finding that it just goes on and on and on.
May 20, 2011