asymptote

American Heritage® Dictionary of the English Language, Fourth Edition

1. n. A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve.

Century Dictionary and Cyclopedia

1. In mathematics, approaching indefinitely close, as a line to a curve, but never meeting. See II.
2. 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. The asymptote is often defined as the tangent to the curve at an infinite distance, and this definition answers for Euclidean space; but, in view of non-Euclidean hypotheses, it is preferable to define it as a common chord of the curve and the absolute (which see), and thus as not necessarily a tangent.
3. 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. The smaller of the surfaces bounded by these lines occupies about one fourth of the circumference of the aperture and is marked with loop-lines of growth curving forward.

Wiktionary

1. n. analysis A straight line which a curve approaches arbitrarily closely, as they go to infinity. The limit of the curve, its tangent "at infinity".
2. n. by extension, figuratively Anything which comes near to but never meets something else.
3. v. analysis To approach, but never quite touch, a straight line, as something goes to infinity.

GNU Webster's 1913

1. n. (Math.) 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.

WordNet 3.0

1. n. a straight line that is the limiting value of a curve; can be considered as tangent at infinity