. Differential and integral calculus, an introductory course for colleges and engineering schools. riting tanMz^j = oo,tan(±-^n =00, etc. It may well be that the student has himself fallen into the errorof giving a too literal interpretation to these equations. For, 7T 7T tan- =00 means, not that tan^ actually exists and has a definitevalue designated by the character 00, but merely that tan x is IT indefinitely large when x is indefinitely near to the value — • Weshall return to this point again in Art. 17. /v>2 — /^2 (5) The function x — a has no value when x = a. For, in that sin x 0 cas
. Differential and integral calculus, an introductory course for colleges and engineering schools. riting tanMz^j = oo,tan(±-^n =00, etc. It may well be that the student has himself fallen into the errorof giving a too literal interpretation to these equations. For, 7T 7T tan- =00 means, not that tan^ actually exists and has a definitevalue designated by the character 00, but merely that tan x is IT indefinitely large when x is indefinitely near to the value — • Weshall return to this point again in Art. 17. /v>2 — /^2 (5) The function x — a has no value when x = a. For, in that sin x 0 case, this function, like , takes the form -, which has no value. x 0 The function has a discontinuity when x = a. Its graph may bedetermined in the following way. 14 DIFFERENTIAL CALCULUS §8 We have the identity (Art. 4) = x + a, which holds true for all values of x except for x = a (see page 5,footnote). The reason that we can assert that this identity fails when x = a is that inthat case the first memberof the identity becomes non-existent, while the secondmember takes the value 2 mX Hence the graph of *2 _ IS x — a identical with the graph of x + a with the single exception that the former lacks the point (a, 2 a). The graph of x + a is the right line shown in the figure, °2 — n^ is this same right line with the point and the graph of(a, 2 a) lacking. x — a (6) A part of the graph of the function 2X is shown in the figure. There is a discontinuity at the origin because the function takes ithere the form 2 °, which hasno meaning. The equation of this curve is of courseiy = &. Now when x is + and very ismall, 2X or y is + and verylarge, and the smaller x becomes,the larger does y become. On the other hand, when x is — and ivery small, 2X or y is + and also very small, and the smaller xbecomes, the smaller does y become. The left-hand branch of thecurve terminates abruptly at the origin. It is plain that the curveis broken at x = 0, and that the severed en
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
