Ex. 5. Evaluate , when x = a. e* — ea m = g^ = g, when* = «, _ /'(*) _ nix — a)"-1 ~ 4>\x) ~ T*' which, when x = a, is either 0, e~a, or 00, according as n is greater than, equal to, or less than, 1. On examination of the above examples, it will be seen that the result is infinitesimal, finite or infinite, according as the order of infinitesimals in the numerator, determined by the method of Section 1 of this Chapter, is higher than, the same as, or lower than, the order of the denominator. 125.] Evaluation of indeterminate quantities of the form ~. Let f(x) and <f> (x) be two functions of x, which become infinite when x = x0; then, as their reciprocals become infinitesimals, the ratio of the functions may be evaluated by the former method, as follows; ^{' - = |§-, when x = x0; f"(x) »(»-l)(»-2)... 3.2.1 . , = = — x = 0, when # = 00 . The result of which shews that ex is, when x = 00, an infinity of a higher order than x"; also a similar property is true of a"; a finite quantity therefore raised to an infinite power is an infinity of a higher order than the same infinite quantity raised to a finite power. € Ex. 4. Evaluate —5—, when x — 0. Replace x2 by i; so that z = 00, when x = 0; and the function becomes = H, when xr = oo; _ w-?"-1 _ 00 - et - 00' n(»-l)...3.2.1 _ , = —i = 0, when z — 00: ez therefore the function is equal to 0, when x = 0; whence we conclude that e~«* is an infinitesimal of a higher order than x2n, however large be the value of n. This is a result of some importance; insomuch as it shews that although e *' vanishes, when x = 0, whence it might be inferred that x is a factor of it, yet it admits of no factor of the form xtn however large n be. Compare equation (23), Art. 116, and the remarks thereupon. Ex. 5. Evaluate ^ —, when x = 00 . x if x therefore becomes infinite, it i9 an infinity of a higher order than its logarithm. 126.] Evaluation of functions which for particular values assume the indeterminate form 0 x oo . Let f(x) and ^>(x) be two functions of x which respectively become 0 and oo , when x = x0; then their product may be put under the form ■ffi , which = when x = ,r0, J(xj and may be evaluated according to the method of the last two Articles. Ex. 1. Evaluate e~x log .r, when x = oo . , log.r _ , e~xiogx = ° = ^, when x = oo; —!— = 0, when x = oc: xe* .■. e~x log = 0, when x = oo . Ex. 2. Evaluate a?" log x, when # = 0. , log a? 1 xn _ . ,r" log # = —— = = = 0, when x = 0: .•. «r"log,r = 0, when x = 0. 127.] Evaluation of functions which for particular values assume the form oo — oo . Let f(x) and <f>(x) be two functions of x, which = 0, when x = x0; in which case, ,, v and ■ 7 . both become 00 : then '/(*) 1 1 0(a?„) -/(a?0) _ 0 . Section 3.—Evaluation of Explicit Functions of a tingle variable, which for particular values assume the forms 0°, oo°, Is0, 0". 128.] Let f(x) and <p(x) be two functions of a? which, when x = x0, assume such values that f(xo)*^ is of one or other of the above forms. Let y =/(*)*(->; .-. logy = <f>(x) log/Or); and as log/(;r) has singular values when f(x) = 0, or = 1, or = 00 , we may express the last equation in the form 1 log/(,r) logy = V , which, for the particular value of x0, will be of the form ^ or ||-, and may be evaluated according to the methods explained in the last Section. Ex. 1. Evaluate xx, when x = 0. Let y = x*, .-. logy = x\o%x; |