95 steadily increasing or decreasing functions if

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95. Steadily increasing or decreasing functions. If there is a number δ such that φ ( x 0 ) 5 φ ( x 00 ) whenever a - δ < x 0 < x 00 < a + δ , then φ ( x ) will be said to increase steadily in the neighbourhood of x = a . Suppose first that x < a , and put y = 1 / ( a - x ). Then y → ∞ as x a - 0, and φ ( x ) = ψ ( y ) is a steadily increasing function of y , never greater than φ ( a ). It follows from § 92 that φ ( x ) tends to a limit not greater than φ ( a ). We shall write lim x a +0 φ ( x ) = φ ( a + 0) . We can define φ ( a - 0) in a similar manner; and it is clear that φ ( a - 0) 5 φ ( a ) 5 φ ( a + 0) . It is obvious that similar considerations may be applied to decreasing functions. If φ ( x 0 ) < φ ( x 00 ), the possibility of equality being excluded, whenever a - δ < x 0 < x 00 < a + δ , then φ ( x ) will be said to be steadily increasing in the stricter sense .
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[V : 96] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 204 96. Limits of indetermination and the principle of conver- gence. All of the argument of §§ 80 84 may be applied to functions of a con- tinuous variable x which tends to a limit a . In particular, if φ ( x ) is bounded in an interval including a ( i.e. if we can find δ , H , and K so that H < φ ( x ) < K when a - δ 5 x 5 a + δ ). * then we can define λ and Λ, the lower and up- per limits of indetermination of φ ( x ) as x a , and prove that the necessary and sufficient condition that φ ( x ) l as x a is that λ = Λ = l . We can also establish the analogue of the principle of convergence, i.e. prove that the necessary and sufficient condition that φ ( x ) should tend to a limit as x a is that, when is given, we can choose δ ( ) so that | φ ( x 2 ) - φ ( x 1 ) | < when 0 < | x 2 - a | < | x 1 - a | 5 δ ( ). Examples XXXV. 1. If φ ( x ) l, ψ ( x ) l 0 , as x a , then φ ( x ) + ψ ( x ) l + l 0 , φ ( x ) ψ ( x ) ll 0 , and φ ( x ) ( x ) l/l 0 , unless in the last case l 0 = 0. [We saw in § 91 that the theorems of Ch. IV , §§ 63 et seq. hold also for functions of x when x → ∞ or x → -∞ . By putting x = 1 /y we may extend them to functions of y , when y 0, and by putting y = z - a to functions of z , when z a . The reader should however try to prove them directly from the formal def- inition given above. Thus, in order to obtain a strict direct proof of the first result he need only take the proof of Theorem I of § 63 and write throughout x for n , a for and 0 < | x - a | 5 δ for n = n 0 .] 2. If m is a positive integer then x m 0 as x 0. 3. If m is a negative integer then x m + as x +0, while x m → -∞ or x m + as x → - 0, according as m is odd or even. If m = 0 then x m = 1 and x m 1. 4. lim x 0 ( a + bx + cx 2 + · · · + kx m ) = a . 5. lim x 0 { ( a + bx + · · · + kx m ) / ( α + βx + · · · + κx μ ) } = a/α , unless α = 0. If α = 0 and a 6 = 0, β 6 = 0, then the function tends to + or -∞ , as x +0, * For some further discussion of the notion of a function bounded in an interval see § 102 .
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[V : 97] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 205 according as a and β have like or unlike signs; the case is reversed if x → - 0.
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