250 DIFFEB.h]NTi`;AIx Game 1 rw _ then the equation of the asymptote is found by subetitu i Q ; ing values a arrl b in the equation Z Z = a + I2 1' If only one of these limits exists, but I limit (%> = M, then we have one intercept and the slope given, so that the equation of the asymptote is y=m:r+b, or z=;Z+a. Itnusrnnxve Exnmna 1. Find the asymptotes to the hyperbole Z; - g: = 1. . dl/ biz b 1 limit (dll) il _ f = 7 = _ -l d = _ = _ . Solution dz asv i a az 1 an m Z = w dx 3; a "5 2 Also z;= LL and yi:-Z; hence these intercepts are zero when a:=y=co. 33 I/ Therefore the asymptotes pass through the origin (see figure on p. 240) and their equations are 0 4; (z Ox), or ay/ ibm This method is frequently too complicated to be of practical use. The most convenient method of determining the asymptotes to algo braic curves is given in the next section 152. Method of determining asymptotes to algebraic curves. Given the algebraic equation in two variables, (A) f(1~ 9) = 0 If this equation when cleared of fractions and radicals is of degree n, then it may be arranged according to descending powers of one of the variables, say y, in tl\e form (B) a_y"+(b:c+c)y"'l+(dx”+e:c+f)y”"+-~-= ."‘ For a given value of rc this equation determines in general n values of y.
- For use in this section the attention of the student is culled to the following theorem
from Algebra: Given an algebraic equation of degree n, Ay»\+ By"-1 + Oy”-2+Dy--8+ When A approaches zero, one root (value of V) approaches oo When A and B approach zero, two roots approach an When A, B, and C approach zero, three roots approach ao, em me 4 ' f/f`i0SOff ® 71- = Q -- = ,Ana a 0 <--==0.
