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

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

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If ε is any given positive number, there exists N, such that | yn \ < ejM for n > N. -^forn> N. Hence, \xn yn \ < ε for n > N, so that xy -> 0. The proof is analogous for non-enumerated variables. We note that the second property is all the more readily justified if a; is a constant. e. the product of a constant and an infinitesimal is an infinitesimal. In view of the fundamental importance of the concept of infinitesimal for what follows, we shall pause to add some remarks supplementary to the above.

This corresponds to taking only positive values of the square root. 34 FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS 21] I t may also be noted t h a t the part of the graph of the function y = ^χ, shown in Fig. 27, is obtained from the part of the graph of the direct function y = x2 (Fig. 13) lying on the right of axis OY. The part of the graph of y =yx or y = x lying in the first quadrant of the axes, has already been illustrated in Fig. 22. We now t u r n to the case when rotation of a single-valued direct function leads to a single-valued inverse function.

Since ax is always positive, the graph is evidently FIG. 29 located above the axis OX, It may be noted further, that the graph of the functiony=(l/a) x can be obtained from the graph of y = αχ, by turning the figure through 180° about the axis 0 7 . This follows directly from the fact that, with such turning, a; becomes —x, and a~x = (1/a) . A further remark: if a = 1, then y = l x , so that we have y = 1 for all x [12]. A logarithmic function is defined by the equation: y~\ogax. ( 16 ) By definition of logarithm, function (16) is the inverse of function <15).

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