Basic Mathematics for the Biological and Social Sciences by F. H. C. Marriott (Auth.)

By F. H. C. Marriott (Auth.)

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The cases of lim/(x) that have been discussed have been those x-+a for which f(a) is undefined. A continuous function has already been mentioned; it means, roughly, a function y =f{x) that is represented graphically by a continuous line. It is easy to see that LIMITS AND CONVERGENCE 57 for a function of this sort, lim/(x) =f(a). This, in fact, is the x-+a analytical definition of a continuous function. For example, the function y = 1, x > 0, y = 0, Λ; < 0 has a discontinuity at x = 0 (Fig. 1). The function is defined at x = 0, but lim/(x) does not exist.

Two simultaneous equations of this form represent (unless the planes are parallel) a line. The neatest way of writing the equation of a line is x — a _y — b _ z — c h U 4 This line obviously passes through (a, b, c), and the coefficients k, h-> h determine its direction. These coefficients are called the direction ratios. In fact, the /'s are proportional to the cosines of the angles the line makes with the coordinate axes. For consider x/lx = y\l% = z/l3, parallel to the line and passing through the origin.

The simplest way of generating a cone from an equation in x and y is to replace x and y by kx/z and ky/z. This represents a cone with its apex at the origin, and its section in the plane z = k is the original curve. So any homogeneous equation—one which is unaltered by multiplying x, y and z by a constant—is a cone with its apex at the origin. For example, x2 + y2 = a2z2 is a right circular cone. The equation y = z sin (x/z) is also a cone in this sense—it is actually a sort of corrugated fan, made up of lines all passing through a single point.

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