Elements of the Differential and Integral Calculus/Chapter XIX

. An asimptote to a curve is the limiting

position* of a tangent whose point of contact moves off an

infinite distance from the origin.

Thus, in the hyperbola, the asymptote

AB is the limitng position of the tangent

PT as the point of contact P moves off to the right to an infinite distance. In the case of algebraic curves the folowing definition is useful: as asymptote is the limiting potition of a secant as two points of intesection of the secant with a branch of the curve move off in the same direction along that branch to an infinite distance. For example, the asymptote AB is the limiting position of the secant PQ as P and Q move upwards to an infinite distance. . The equation of the tangent to a curve at (x1,y1) is, by (1), p.76,