. Differential and integral calculus, an introductory course for colleges and engineering schools. tothe value — 1. For when x approaches 2 from the left (, remains l 3-2 always less than 2), the exponent of 2X remains always negative and l—s 4 2 decreases without limit, and so f(x) approaches the value - — 1, or 1. A On the other hand, if x approaches 2 from the right, the exponent of 62 DIFFERENTIAL CALCULUS §48 2 remains positive and 2X 2 increases without limit, and j(x) ap- i_ x — proaches the value — — 1, or — 1. When x = 2, 2 may be regarded as nonexistent, but f(x) exists, and has
. Differential and integral calculus, an introductory course for colleges and engineering schools. tothe value — 1. For when x approaches 2 from the left (, remains l 3-2 always less than 2), the exponent of 2X remains always negative and l—s 4 2 decreases without limit, and so f(x) approaches the value - — 1, or 1. A On the other hand, if x approaches 2 from the right, the exponent of 62 DIFFERENTIAL CALCULUS §48 2 remains positive and 2X 2 increases without limit, and j(x) ap- i_ x — proaches the value — — 1, or — 1. When x = 2, 2 may be regarded as nonexistent, but f(x) exists, and has the value +1. The student should calculate a few values off(x) by the aid of a table of loga-rithms, say for x = , ,, ,, Discontinuities like those justdescribed will not be consideredfurther because very few of thefunctions dealt with in this bookhave such discontinuities. Thefunctions to be met with in thefollowing pages are for the mostpart such as are continuousthroughout their whole extent,or such as become discontinuous through taking the forms ~ or oo,. like smz ,2 _ and tan x. x x — aObserve that a function nei-ther ceases to exist nor becomesdiscontinuous merely by taking complex values. For example,Vx2 — 1 is imaginary for all values of x between —1 and +1, butis nowhere nonexistent or discontinuous. The rigorous analytical definition of continuity is as follows:If f(a) exists, and if lim f{x) = f(a), no matter in what way x x=a approaches its limit a, f(x) is defined to be continuous at a. If, on the other hand, f(a) does not exist, or if, when f(d) does existx can be made to approach a in such a way that lim f(x) y£ f{a), f(x) x=a is defined to be discontinuous at a. It can be shown that every case of discontinuity thus far men-tioned is covered by this definition. §§49-50 SOME GENERAL PROPERTIES OF FUNCTIONS 63 49. Discontinuities of f(x). A function may itself be con-tinuous within an interval and yet have a derivative w
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