Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/273

CHAPTER XIX ASYMPTOTES. SINGULAR POINTS 150. Rectilinear asymptotes. An asymptote to e curve is the limiting position* of a tangent whose point of contact moves off to an infinite distance from the origin. Thus, in the hyperbola, the asymptote AB is the limiting position of the tangent PT as the point of contact P moves off to the right to an intinite distance. In the case of algebraic curves the following definition is useful: an asymptote is the limiting position of a secant as two points of intersection 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. 151. Asymptotes found by method of limiting intercepts. The equation of the tangent to a curve at (xl,yi) is, by (1),p.76, dy y-@/.=§;e-w,>- Q First placing y = 0 and solving for ar, and then placing x = 0 and solving for y, and denoting the intercepts by :ai and yi respectively, we get ,1 xi: xl- yx # = intercept on OX; l yi = y‘- x|% = intercept on OK Since an asyinptote must pass within a. finite distance of the origin, one or both of these intercepts must approach iinite values as limits when the point of contact (11, yi) moves off to an infinite distance. If limit (z,) = a and limit (yi) = b, ‘A line that approaches a fixed straight line as a limiting position cannot be wholly at iniinity; hence it lolluws that an nsymplote must pass within a finite distance of the origin. It is evident that n curve which has no infinite branch can hiwe no real asymptote. fOr, less precisely, an asympwte to a curve is sometimes denned as a tangent whose point ot contact is at an infinite distance. 249 1