PureMath.pdf

# The proofs of the inequalities which are used here

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The proofs of the inequalities which are used here depend on certain properties of the area of a sector of a circle which are usually taken as geometrically intuitive; for example, that the area of the sector is greater than that of the triangle inscribed in the sector. The justification of these assumptions must be postponed to Ch. VII .

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[V : 98] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 210 14. lim 1 - cos x x 2 = 1 2 . 15. lim sin αx x = α . Is this true if α = 0? 16. lim arc sin x x = 1. [Put x = sin y .] 17. lim tan αx x = α , lim arc tan αx x = α . 18. lim cosec x - cot x x = 1 2 . 19. lim x 1 1 + cos πx tan 2 πx = 1 2 . 20. How do the functions sin(1 /x ), (1 /x ) sin(1 /x ), x sin(1 /x ) behave as x 0? [The first oscillates finitely, the second infinitely, the third tends to the limit 0. None is defined when x = 0. See Exs. xv . 6, 7, 8.] 21. Does the function y = sin 1 x sin 1 x tend to a limit as x tends to 0? [ No . The function is equal to 1 except when sin(1 /x ) = 0; i.e. when x = 1 , 1 / 2 π , . . . , - 1 , - 1 / 2 π , . . . . For these values the formula for y assumes the meaningless form 0 / 0, and y is therefore not defined for an infinity of values of x near x = 0.] 22. Prove that if m is any integer then [ x ] m and x - [ x ] 0 as x m +0, and [ x ] m - 1, x - [ x ] 1 as x m - 0. 98. Continuous functions of a real variable. The reader has no doubt some idea as to what is meant by a continuous curve . Thus he would call the curve C in Fig. 29 continuous, the curve C 0 generally continuous but discontinuous for x = ξ 0 and x = ξ 00 . Either of these curves may be regarded as the graph of a function φ ( x ). It is natural to call a function continuous if its graph is a continuous curve, and otherwise discontinuous. Let us take this as a provisional definition and try to distinguish more precisely some of the properties which are involved in it. In the first place it is evident that the property of the function y = φ ( x ) of which C is the graph may be analysed into some property possessed by
[V : 98] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 211 X Y ξ 0 ξ ′′ ξ 1 ξ 2 ξ C C C C P Fig. 29. the curve at each of its points. To be able to define continuity for all values of x we must first define continuity for any particular value of x . Let us therefore fix on some particular value of x , say the value x = ξ corresponding to the point P of the graph. What are the characteristic properties of φ ( x ) associated with this value of x ? In the first place φ ( x ) is defined for x = ξ . This is obviously essential. If φ ( ξ ) were not defined there would be a point missing from the curve. Secondly φ ( x ) is defined for all values of x near x = ξ ; i.e. we can find an interval, including x = ξ in its interior, for all points of which φ ( x ) is defined. Thirdly if x approaches the value ξ from either side then φ ( x ) ap- proaches the limit φ ( ξ ).

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