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Let I (a, b) be a bounded open interval and f: I R be a monotone increasing function on I. (i) If f is bounded above on I, then lim f(x) = sup f(x). xE (a,b) %3D (ii) If f is bounded below on I, lim f(x) = x a+ inf f(x). xE (a,b) then

Let I (a, b) be a bounded open interval and f: I R be a monotone increasing function on I. (i) If f is bounded above on I, then lim f(x) = sup f(x). xE (a,b) %3D (ii) If f is bounded below on I, lim f(x) = x a+ inf f(x). xE (a,b) then

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove. ii.

Let I = (a, 6) be a bounded open interval and f :I → %3D R be a monotone increasing function on I. (i) If f is bounded above on I, then lim f(x) sup f(x). πε (α,) x→b- (ii) If f is bounded below on I, then lim f(x) : inf f(x). xE (a,b) x-- a+ (iii) If f is unbounded above on I, then lirn f(x) (iv) If f is unbounded below on I, then lim f (x) X -- a+
Transcribed Image Text:Let I = (a, 6) be a bounded open interval and f :I → %3D R be a monotone increasing function on I. (i) If f is bounded above on I, then lim f(x) sup f(x). πε (α,) x→b- (ii) If f is bounded below on I, then lim f(x) : inf f(x). xE (a,b) x-- a+ (iii) If f is unbounded above on I, then lirn f(x) (iv) If f is unbounded below on I, then lim f (x) X -- a+
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