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 shall consider the function: y = arc sin x, (22) 42 FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS 24] The graph of this function (Fig. 39) is obtained from the graph of y = sin x by the rule given in [20]. e. the function (22) is only defined in the interval — 1 < x < -f- 1. Furthermore, equation (22) is equivalent to the equation sin y = x; and, as is known from trigonometry, for a given x we obtain an infinite number of values of y. We see from the graph, in fact, t h a t perpendiculars to the axis OX from points in the interval FIG.

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. 3. The limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is not zero.

Let the integer n increase indefinitely in equation (7). e. lim - ^ - = 0. e. equation (8) is valid for any given x, positive or negative. We obtain the limit in this example, after first showing that it exists. If we did not show its existence, our method could lead to a false result. Consider, for instance, the sequence: Ul = q, u2 = q2, . . 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].

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