22 INFINITESIMAL CALCULUS We sha the true But 11 first consider the case whero/() = 0, and c(a) = 0. Here alue of is that of wh^Ti7> . isnvmirsfp.nt,. <j>(a) tf>(a + h.) f(a + It) /() + Jif (a + eh) f (a + Gli) ./J! i hen7t = 0. Hence the limiting value of the fraction is in this case represented by / ()_ w* Aain, if * ^ a be also of the form , its true value is that of ^ <t(a)
- ^ n ; and soon.
<p"(a) In general, if the order of the lowest derived functions which do not both vanish is n, then the true value of &L is that of I H <t>(a) <f>("a) sin ay - r si For example, the fraction tan ay - tan ax , to find its true value. is o f the form
when x Here f(x) = x a sin ay - y a sin ace, <f>(x) = tan ay - tan ax , , *, / ^) === &3 ~ sin cty cty cos o/x ^ <p ^*J == ^ sec** accordingly the required value is represented by i-^-11- = y a l (y cos y ~ ?i n a l/) cos 2 7/ . 59. Again, to find the true value of^-;, if /(a) = 00, <() = co <() Here a! . . . . , , . , , ^~ , which is of the form -/r , when x=a. Hence, by the former case, its limiting value is that of -!} 1 2 (*) ) Suppose A to represent the limiting value in question and we have A = ^77 r A , or A = ,. -r- Accordingly the true value of the indeterminate form - is found
in the same manner as that of the form
In the preceding, in dividing both sides of our equation by A, we have assumed that A is neither zero nor infinite. It can, however, be easily shown that the true value in. either of these cases is still that of ; . 60. Again, the expression f(x) x tf>(x) becomes indeterminate for any value of x which makes one of its factors zero and the other infinite. The expression, howeyer, is readily reduced to the form
- for, if/() = 0, and $(a) = oo, we have
/()-?- - -r- , which is of the form 4>(a) /(a) x < Also, if the true value of be unity when ) = oo, then This is of the form , and its true value can in general be found as above. By this means the true value of f(x) <(>(x) when f(x) = oo , and 0(,r) = oo can be found. 61. The expression U* becomes indeterminate in some cases ;-for suppose j/ = ?t r , then log y = v lo This latter product becomes indeterminate whenever one of its factors is zero a7id the other infinite. (1) Let v = Q, and log w=oo ; the latter equation requires either u = a> , or = 0. Consequently u v becomes indeterminate for either of the forms or oo . (2) Let i>=oo, log w = 0; the latter equation gives = 1, and the corresponding indeterminate forms are I 00 or 1-". 62 . In many cases the true value of an indeterminate form can be best determined by ordinary algebra or trigonometry. Thus, for example, the expression Vff a + ax + x* - X/a? - ax - x a is of the form when x = 0. Va - x To find its true value we multiply the true value of which is plainly /a, when a;=0. 63. The differential calculus was applied for the first time to find ing the true value of an indeterminate form by John Bernoulli, in the Acta Eruditorum, 1704, when studying the problem of drawing the tangents at a multiple point on a curve. This problem, as stated already in the Introduction, was started by Rolle, as a crux for the advocates of the differential calculus. It may be here remarked that the determination of the tangents at a multiple point is gene rally much simpler by Cartesian coordinate geometry than by the method of the differential calculus. A few elementary examples are added of the different classes of indeterminate forms here given. when x = a . yKj == ? <% (t) * 3 ; when r = l JJg|Q f(% = flHX *< Accordingly, when r>l , u = (2) For (3) Since when
- "^ ;andforr<l ,
8 + 2 cos x - 2 , when x . tan x the same as that of = 1 when x = Q, the true value of u, in this case, is i o , and is easily seen to be A- (4) = , when a;=oo. Here u = / ~ ; but the true value of , when a; = oo , is easily seen to be zero, consequently the true value of u is also zero. (5) = ( 1 + | j (1) wnen x = Q, and (2) when x = <x>.
x )
The true values are (1) u = l ; (2) u^c a . (6) V^ J + ax- Vx 2 + te, when a; = QO . This is of the form oo - oo ; its true value, however, is that of a-b a-b a 2 + ax + /x z + bx / a / V 1+ 1T + V a;" -i/ n = _ sna; (9) This is equivalent to value is - log a. _ (10) V-g-^sing ] ^ x(l -a x ), when a? = 00 . ~ a when 2 = 0, and accordingly its true when a? = True value, Maxima and Minima. 64. "We have seen in the Introduction that the question of finding the greatest and least values of an expression was, in the hands of Fermat, one of the first applications of the method of infinitesimals. We have also seen that the principle of his method had been pre viously stated correctly by Kepler, and is the same as that obtained by the differential calculus. We now proceed to a more general in vestigation on maxima and minima. Let u represent the function, and x the variable, and suppose we have ii=f(x). Let a be a value of x corresponding to a maximum or a minimum value of u, then for a maximum we must have /(a) >f(a + 7t), and f(a) >f(a -h), for small values of h ; and for a minimum, f(a)<f(a + k), and/()</(a-A) . A ccordingly, in either case, /( + 7t) -/() and f(a - h) -f(a) must have the same sign, h being small.
