We accordingly frame the following formal definition

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We accordingly frame the following formal definition: Definition I. The function φ ( n ) is said to tend to the limit l as n tends to , if, however small be the positive number , φ ( n ) differs from l by less than for sufficiently large values of n ; that is to say if, however small be the positive number , we can determine a number n 0 ( ) corresponding to , such that φ ( n ) differs from l by less than for all values of n greater than or equal to n 0 ( ) . It is usual to denote the difference between φ ( n ) and l , taken positively, by | φ ( n ) - l | . It is equal to φ ( n ) - l or to l - φ ( n ), whichever is positive, and agrees with the definition of the modulus of φ ( n ) - l , as given in Chap. III , though at present we are only considering real values, positive or negative. With this notation the definition may be stated more shortly as follows: if, given any positive number, , however small, we can find n 0 ( ) so that | φ ( n ) - l | < when n = n 0 ( ) , then we say that φ ( n ) tends to the limit l as n tends to , and write lim n →∞ φ ( n ) = l ’. Sometimes we may omit the ‘ n → ∞ ’; and sometimes it is convenient, for brevity, to write φ ( n ) l . The reader will find it instructive to work out, in a few simple cases, the explicit expression of n 0 as a function of . Thus if φ ( n ) = 1 /n then l = 0, and the condition reduces to 1 /n < for n = n 0 , which is satisfied if n 0 = 1 + [1 / ]. * There is one and only one case in which the same n 0 will do for all values of . If, from a certain value N of n onwards, φ ( n ) is constant, say equal to C , then it is evident that φ ( n ) - C = 0 for n = N , so that the inequality | φ ( n ) - C | < is satisfied for n = N and all positive values of . And if | φ ( n ) - l | < for n = N and all positive values of , then it is evident that φ ( n ) = l when n = N , so that φ ( n ) is constant for all such values of n . * Here and henceforward we shall use [ x ] in the sense of Chap. II , i.e. as the greatest integer not greater than x .
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[IV : 60] LIMITS OF FUNCTIONS OF A 140 59. The definition of a limit may be illustrated geometrically as fol- lows. The graph of φ ( n ) consists of a number of points corresponding to the values n = 1, 2, 3, . . . . Draw the line y = l , and the parallel lines y = l - , y = l + at distance from it. Then lim n →∞ φ ( n ) = l, if, when once these lines have been drawn, no matter how close they may 1 2 3 O X Y n 0 y = l - ǫ y = l + ǫ Fig. 27. be together, we can always draw a line x = n 0 , as in the figure, in such a way that the point of the graph on this line, and all points to the right of it, lie between them. We shall find this geometrical way of looking at our definition particularly useful when we come to deal with functions defined for all values of a real variable and not merely for positive integral values. 60. So much for functions of n which tend to a limit as n tends to . We must now frame corresponding definitions for functions which, like the functions n 2 or - n 2 , tend to positive or negative infinity. The reader should by now find no difficulty in appreciating the point of Definition II. The function φ ( n ) is said to tend to + (positive infinity) with n , if, when any number Δ , however large, is assigned, we
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[IV : 61]
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