. An elementary course of infinitesimal calculus . n books onTrigonometry. The function sin x is continuous for all values of x. For S (sin x) = sin (x + Sx) — sin x = 2 sin ^Bx. cos (x + ^Sx). The last factor is; always finite, and the product of theremaining factors can be made as small as we please bytaking Sx small enough. In the same way we may shew bhat cos x is result is, however, included in the former, since cos X = sin {x + Itt). ... , sin a; Aijam, since tan x = , ^ cos a; the continuity of sin x and cos x involves (Art. 13) thatof tan X, except for those values of x
. An elementary course of infinitesimal calculus . n books onTrigonometry. The function sin x is continuous for all values of x. For S (sin x) = sin (x + Sx) — sin x = 2 sin ^Bx. cos (x + ^Sx). The last factor is; always finite, and the product of theremaining factors can be made as small as we please bytaking Sx small enough. In the same way we may shew bhat cos x is result is, however, included in the former, since cos X = sin {x + Itt). ... , sin a; Aijam, since tan x = , ^ cos a; the continuity of sin x and cos x involves (Art. 13) thatof tan X, except for those values of x which make cos x = are given by aj = (n + ^) tt, where n is integral. In the same way we might treat the cases of sec x,cosec X, cot X. The figures on p. 84 shew the graphs of sin x and tan reader should observe how immediately such relationsas sin (—«)=— sin x, sin (tt — «) = sin x, sin (a; + tt) = — sin x, tan (a; + tt) = tan x can be read off from the symmetries of the curves. L. 3 84 INFINITESIMAL [CH. 1. Fig. 19. 16-17] CONTINUITY. 35 17. The Exponential Function. We consider next the exponential function. This maybe defined in various ways; perhaps the simplest, for ourpurpose, is to define it as the sum of the infinite series i+^+ri+rl:T3+ (^)- To see that this series is convergent, and has therefore adefinite sum, for any given value of x, we notice that the ratioof the (m + l)th term to the mth is aj/m. This ratio can be madeas small as we please (in absolute value) by taking m great a point in the series can always be found after which thesuccessive terms will diminish more rapidly than those of any givengeornetrical progression whatever. The series is therefore con-vergent, by Art. 6. It is, moreover, absolutely convergent. If we denote the sum of the series (1) by E(x), it may beshewn that E{x)xE{y) = E{x + y) (2). The proof involves the rule for multiplication of absolutelyconvergent series. Let and ?!;„ +Uj + i
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