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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Extra resources for A Course of Higher Mathematics. Volume I

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Then evidently, # - > + °o. If its sequence of values is: —1, —2, —3, . . , then x-> — co. And finally, if the sequence is: —1, 2, —3, 4, . , we can write: x->- co. 2, i ) , (β) and let M be any given positive number. e. the variable in question tends to + °°. 56 FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS [30 If q is replaced b y (-— q) in the sequence (5), the only change is in the signs of odd powers of q, the absolute values of the members of the sequence remaining as before; thus, for negative q, with absolute value greater t h a n unity, the sequence (5) tends to infinity.

Let us take the graph of the direct function y = f(x). I t can easily be seen t h a t the same graph can serve as a graph of the inverse function x = (p{y). I n fact, both the equations y — f(x) and x = q>(y) give the same functional relationship between x and y. Suppose an arbitrary x is given in the direct function. If we mark off an interval from the origin 0 32 FUNCTIONAL BELATIONSHIPS AND THE THEORY OF LIMITS [20 along the axis OX, corresponding to the number x, then erect a perpendicular to OX from the end of this interval as far as its intersection with the graph, we obtain the value of y corresponding to the chosen x as the length of this perpendicular, with the corresponding sign.

The ordinate of this curve increases in absolute value when x increases in absolute value, the increase being the faster, the greater the absolute value of a. Figure 14 shows a series of graphs of the function (5) for different values of a, these values being marked in the figure against the corresponding parabolas. e. if a given point (x, y) lies on the parabola (5), the point (—x, y) also lies on it. e. one of them is the mirror image of the other relative to this axis. Thus, if the right-hand portion of the plane is turned through 180° about the axis OY and combined with the left-hand portion, the part of the parabola lying to the right of the axis OY will coincide with the part of the parabola lying to the left of this axis.

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