. An elementary course of infinitesimal calculus . f(x). If F(x) be of higher degreethan /(«) the ordinates become infinite; if of lower degreethe ordinates diminish indefinitely, the axis of x beingan asymptote; if the degrees are the same, there is anasymptote parallel to x. * It is assumed that the fraction has been reduced to its lowest terms. 14-15] CONTINUITY. 29 In cases where the degree of the numerator is not lessthan that of the denominator, it is convenient to performthe division indicated until the remainder is of lower degreethan the divisor, and so express y as the sum of an inte
. An elementary course of infinitesimal calculus . f(x). If F(x) be of higher degreethan /(«) the ordinates become infinite; if of lower degreethe ordinates diminish indefinitely, the axis of x beingan asymptote; if the degrees are the same, there is anasymptote parallel to x. * It is assumed that the fraction has been reduced to its lowest terms. 14-15] CONTINUITY. 29 In cases where the degree of the numerator is not lessthan that of the denominator, it is convenient to performthe division indicated until the remainder is of lower degreethan the divisor, and so express y as the sum of an integralfunction and a proper fraction. Ex. 1. y = — x_ . 1 This makes y = 0 for a; = 1, and j/ = + oo for x = 0. Also y ispositive for 1 > aj > 0, and negative outside this interval. Fromthe second form of y it appears that for a; = + oo we have y = — J-We further find, as corresponding values of x and y:Cx = -co, -3, -2, -1, --5, 0, -5, 1, 2, 3, +<»,ly = --5,--67, -75, -1, -1-5, +00, -5, 0, --25, --33, figure shews the 30 INFINITESIMAL CALCULUS. [CU. 1 Ex. 2. \+x 2l+x Here y = 0 for x = 0 and a; = 1, aud y = + oo for a; = — 1. Also ychanges sign as x passes through each of these values. Fornumerically large values of x, whether positive or negative, thecurve approximates to the straight line y^-x + 2, lying beneath this line for a; = + oo , and above it for a; = — oo .The figure shews the curve.
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