Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/275

 h e ‘ mmm or  251 CASE I. To determine the asymptotes to the curve (B) which are parallel to the coiirdinate axes. Let us first investigate for asymptotes parallel to 01/I The equation of any such asymptote is of the form (0) @=/C, and it must have t\vo points of intersection with (B) having infinite ordinates. First. Suppose a is not zero in (B), that is, the term in y” is present. Then for any finite value of r, (B) gives n values of y, all fnite. llence all such lines as (C) will intersect (B) in points having finite orclinates, and there are no asymptotes parallel to OK Second. Next suppose a= 0, but I2 and c are not zero. Then we know from Algebra that one root (=_1/) of (B) is infinite for every finite value of ac; that is, any arbitrary line (C) intersects (B) at only one point having an infinite ordinate, If now, in addition, l>z+c=0, or (D) ¢=-g, then the first two terms in (12) will drop out, and hence two of its roots are infinite. That is, (D) and (B) intersect in t\vo points having infinite ordinates, and therefore (D) is the equation of an asymptote to (B) which is parallel to Oli Third. lf a = b = c = 0, there are two values of at that make _y in (B) infinite, namely, those satisfying the equation (E) ` dr” + ez +f= 0. Solving (E) for ac, we get two asymptotes parallel to OK and so on in general. In the same way, by arranging _/`(:c, y) according to descending powers of rc, we may find the asymptotes parallel to OX Hence the following nxle for finding the nsymptotes parallel to the coordinate axis: Fmsr STEP. Eguate to zero the coeficient of the highest power of z in the equation. This gives all asyrnptotes parallel to OX SECOND STEP. Equate to zero the coqfcient of the hrllrhest power of y in the equation. This _#UGS all asymptotes parallel to OK Nou; Qf course if one or both qf these coqfcients do not involve z (or gy), they cannot be zero, and there will be no corresponding asymptote. -