A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)

By V. I. Smirnov and A. J. Lohwater (Auth.)

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We confine ourselves to the case of the product xy of two variables. We suppose t h a t x and y vary simultaneously, tending respectively to limits a and b, and we show t h a t xy tends to the limit ab. We have by hypothesis: x = a + a, y = b + β , 28] 53 BASIC THEOBEMS where a and ß are infinitesimals; hence: xy = (a + a) (b + ß) =ab + {aß + ba + aß). Using both of the properties of infinitesimals from [26], we see t h a t the sum in the bracket on the right of this equation is an infinitesimal, and hence we have: lim (xy) = ab = lim x · lim y.

We write, in fact: m = Obviously: and also: ^1 VAl + Bl ' χ = ^1 VAl + Bi ' A = ]/Al + Bi· -4X = m^4, B1 = nA , (20) m2 + n2 = 1 , \m\ < 1, | n | < 1, so that, from trigonometry, an angle bx can always be found such that: cos b1 = m , sin b1 = n. e. y = A (cos δ χ · sin a±x + sin 6X · cos axx) , y = Asm {ax x -f- δχ) 24· Inverse trigonometric, or circular, functions· These functions are obtained by inversion of the trigonometric functions: y = sin x, cos x, tan x, cot x , their symbols being respectively: y = arc sin x, arc cos #, arc t a n x, arc cot # ; t h e s e symbols are simply abbreviated forms of description for the angle (or arc), of which the sine, cosine, tangent or cotangent is respectively equal to x.

A variable x is said to be bounded, if there exists a positive number M, such that \ x | < M for all values of x. We can take x = sin a as an example of a bounded magnitude, where the angle a varies in any manner. Here, M can be taken as any number greater than unity. We now consider the case when the point K is displaced successively, and indefinitely approaches the origin. More precisely, we suppose 26] INFINITESIMALS 45 t h a t successive displacements of point K bring it inside any previously assigned small section S'S of t h e axis OX with centre 0, and t h a t it remains inside this section on further displacement.

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