Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/276

 _-v _ L 252 DIFFERENTIAL CALCULUS Ir.Lus'r1u'rxvs Exnnns l. Find the ssympwtm of the curve nfs: = plz -ef V Solution. Arranging the terms according I/o powers of an, .- y/zz-(2a1/+a.9)z+a2y=0. Y Equating to zero the coeillcient of the high- est power of z, we get y = 0 as the asyinptote parallel to OX. In fact, the asymptote win- cides with the axis of 1. Arranging the terms according to the powers of y, (a:-a)21/-a“z = 0. E Placing the ooefiicient of 7/ equal to zero, 0 A X we get as = u. twice, showing that AB is a double asymptote parallel to OY. If this curve is examined for asymptotes oblique to the axes by the method explained below, it will be seen that there are none. Hence 7/ = 0 and an = a. are the only asyinpwtes of the given curve. CASE II. To determine ssymptotes oblique to the coordinate axes. Given the algebraic equation (F) f(1'1 .7/) = 0' Consider the straight line (G) y = mx + lc. It is required to determine ni and lc so that the line (G) shall be an ssymptote to the curve (F). Since an asymptote is the limiting position of a secsnt as two points of intersection on the same branch of the curve move oif to an infinite distance, if we eliminate y between (F) and (G), the resulting equa- tion in z, namely, (H)  rrwv+1c)=0, must have two infinite roots. But this requires that the coeilicients of the two highest powers of ar shall vanish. Equating these coeffi- cients to zero, we get two equations from which the required values of rn and lc may be determined.. Substituting these values in (GQ gives the equation of an asymptote. Hence the following rule for Ending asymptotes oblique to the ciiiniinate axes: Fmsr Sree. Replace 3/ by mz+ lc in the _qivem eqfuaiion and expand. Siscorrn STEP. Arrange the terms according to descending powers qfzv. Truim STEP. Eqz/,ate to zero the coqgicien/ts qf tlw two higlwst powers* of 1, and solve for 'IIL and lc. s lf nie term involving nf- -1 is missing, or ir the value nr ni nbuiinnd by planing uni first meilicient equal to zero causes the seoond eoeflicient to vanish, then hy placing the maili- ek-nts of ze and x'--2 equal to zero we obtain two equations from which the values of m and lc may be found. ln this msc we shall, in geneml, obtain two Ic's for each m, that is, pairs of parallel oblique asymptotes. Similarly, if the term in in*-2 is also missing, each mine of in iurnishes three parallel oblique asyinptotes, and no nn. hi/ '1»7mrosoi'i