. Differential and integral calculus, an introductory course for colleges and engineering schools. x is a trifle I less(greater When than ± , tan x is a very large INTRODUCTION 13 j _ | number, so that if we draw the graph of the function it will be seen to consist of an unlimited number of separate this graph are no points whose abscissas are 7T ,3 7T ,57T 2 ~2~ ~2~ The graph is broken at each of these points, and, as in the graphof -, the severed ends are widely sundered. X. The student must not infer from the foregoing discussion thatthe textbooks in trigonometry are in error in
. Differential and integral calculus, an introductory course for colleges and engineering schools. x is a trifle I less(greater When than ± , tan x is a very large INTRODUCTION 13 j _ | number, so that if we draw the graph of the function it will be seen to consist of an unlimited number of separate this graph are no points whose abscissas are 7T ,3 7T ,57T 2 ~2~ ~2~ The graph is broken at each of these points, and, as in the graphof -, the severed ends are widely sundered. X. The student must not infer from the foregoing discussion thatthe textbooks in trigonometry are in error in writing 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 i
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