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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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. I n this case, we say t h a t the magnitude x tends to zero or is an infinitesimal. We denote the length of the interval S'S b y 2ε, where ε signifies any given positive number. If the point K is inside S'S, then OK < ε and conversely, if OK < ε, Κ is inside S'S.

Un = qn, . . {q > 1). We have obviously: ^„ = V i î · We denote the limit of un by u, without troubling about its existence. e. u{\ — q) = 0 , u= 0. But this result is false, since we know that for q > 1, lim qn = = + oo [29]. 31. Cauchy's test for the existence of a limit. The test for the existence of a limit given in [30] is a sufficient, but not a necessary f The symbol n\ called "factorial w", is short for the product 1 · 2 · 3 . n. 58 FUNCTIONAL RELATIONSHIPS AND THE THEORY OP LIMITS [31 condition that a limit exists, since, as we know [27], a variable can tend to a limit without varying monotonically.

Let us consider the graph of a single-valued increasing function, defined in the interval a < x < b (Fig. 28). L e t / ( a ) = c and/(6) = d, so t h a t evidently, since the function is increasing, c < d. e. for every y in the interval c < y < d there is one corresponding definite value of x. I n other words, the inverse of an increasing function is single-valued. I t is clear from the figure t h a t this inverse function is increasing. e. if x2 > xx implies f(x±) > f{x2)· I t c a n ^ e shown, as above, t h a t the inverse of a decreasing function is a single-valued, decreasing function.

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