Asymptote

a straight line approached by a given curve as one of the variables in the equation of the curve approaches infinity.
Historical Examples

But after you have demonstrated to him the properties of the hyperbola and its asymptote, the apparent absurdity vanishes.
The Religion of Geology and Its Connected Sciences Edward Hitchcock

asymptote, a line always approaching some curve but never meeting it.
The Nuttall Encyclopaedia Edited by Rev. James Wood

The curve is symmetrical about the axis of y and has the axis of x for its asymptote.
Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 7 Various

Even so there are asymptote figures in geometry where an infinite length makes only a finite progress in breadth.
Theodicy G. W. Leibniz

If these factors are all real and distinct, there is an asymptote corresponding to each factor.
Encyclopaedia Britannica, 11th Edition, Volume 14, Slice 5 Various

The area of the loop, which equals the area between the curve and its asymptote, is 3a/2.
Encyclopaedia Britannica, 11th Edition, Volume 10, Slice 5 Various

noun
a straight line that is closely approached by a plane curve so that the perpendicular distance between them decreases to zero as the distance from the origin increases to infinity
n.

“straight line continually approaching but never meeting a curve,” 1650s, from Greek asymptotos “not falling together,” from a- “not” + syn “with” + ptotos “fallen,” verbal adjective from piptein “to fall” (see symptom). Related: Asymptotic.
asymptote
(ās’ĭm-tōt’)

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