. Differential and integral calculus, an introductory course for colleges and engineering schools. come to the value 1, so that the curve approaches as near as weplease to the point (0, 1) from either side of the y-axis. The point(0, 1), however, does not itself belong to the graph: the graphmay be regarded as broken in two at this point or as having had ahole punched in it here. This point is a singular point, or a pointof discontinuity of the graph, or of the function. (2) The function — has no existence when x = 0. For when x = 0, the function has the form - , which has no value because it
. Differential and integral calculus, an introductory course for colleges and engineering schools. come to the value 1, so that the curve approaches as near as weplease to the point (0, 1) from either side of the y-axis. The point(0, 1), however, does not itself belong to the graph: the graphmay be regarded as broken in two at this point or as having had ahole punched in it here. This point is a singular point, or a pointof discontinuity of the graph, or of the function. (2) The function — has no existence when x = 0. For when x = 0, the function has the form - , which has no value because it is impossible to divide 1 by function has, therefore, adiscontinuity when x = 0, buthas a determinate value for allother values of x. The graphis shown in the figure. It hasno point whose abscissa is 0,that is, it has no point in com-mon with the 2/-axis. When x. is a very small + or number, — is a very large + number, and X1 12 DIFFERENTIAL CALCULUS §8 the smaller x becomes the larger is -5 . We majr say, then, that the curve is broken in two at the point whose abscissa is 0, butthat, unlike the preceding curve, the break has receded along the2/-axis to an indefinitely great distance from the origin. We mayhere introduce the point at infinity or the infinite point of a line,which is a fictitious or ideal point whose distance from any finitepoint of the line is greater than any magnitude that can be may then say that the infinite point of the ?/-axis is a point of discontinuity of the graph of -= or of the function 1_ x1 (3) The function - is discontinuous when xx 1 0 because ^ has no value. The graph of this function is an equilateral hyperbola having the axes of coordinatesfor asymptotes as shown in thefigure. The graph has no pointwhose abscissa is 0, but when x — X is a very small =L number, - is x a very large ± number. The
Size: 1870px × 1336px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
