By V. I. Smirnov and A. J. Lohwater (Auth.)
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Extra info for A Course of Higher Mathematics. Volume I
Finally, Fig. 37 gives the graph of y = cot x, which is also made up of separate infinite branches. FIG. e. sin (x i 2π) = sin x and cos (x ± 2π) = cos x, 40 FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS [23 for any x. The graphs of functions y = tan x and y = cot x similarly coincide on displacement along the axis OX by the amount π. The graphs of the functions: y — A sin ax, y = A cos ax (A > 0, a > 0) (17) are always similar to those of y = sin x and y = cos x. For example, to obtain the graph of the first of functions (17) from the graph of y = sin x, the lengths of all the ordinates of this latter function must be multiplied by A, and the scale on the axis OX changed in such FIG.
E. the line joining them is bisected at O. It follows from this that every chord of the curve (8) that passes through the origin of coordinates O is bisected there. In other words: the origin of coordinates O is the centre of the curve (8). A further particular case of a third degree parabola will be mentioned: y = ax2 + ex. (9) 18] THE LAW OF INVERSE PROPORTIONALITY 27 The right-hand side of this equation is the sum of two terms, and consequently, to construct the curve, it is sufficient to draw the straight line y = ex (10) and take the sums of corresponding ordinates of graphs (8) and (10) directly from the figure.
Conversely, if a variable x can be expressed in the form (2), where a is a constant, and a is an infinitesimal, the difference x — a will be an infinitesimal, and hence, a is the limit of x. If the sequence xv x2, . . tends to the limit a, every infinite subsequence xni, xn2, . . contained in the first sequence, also tends to a. I n this subsequence, the subscript nk increases with increasing 1c and runs through some part of the set of positive integers. There is no analogous property, generally speaking, for a non-enumerated variable tending to a limit.
A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)