Limits - Limits and Continuity Definitions of Limits x ! a...

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Unformatted text preview: Limits and Continuity Definitions of Limits x ! a lim f ( x ) = L . For all ! > there is a ! > such that for all x with x ! a < " and with x ! a , f ( x ) ! L < " . n ¡¢ lim a n = L For all ! > there is an N such that for all n ! N , a n ! L < " . x ! a + lim f ( x ) = L For all ! > there is a ! > such that for all x with < x ! a < " , f ( x ) ! L < " . x ¡ a ¢ lim f ( x ) = L For all ! > there is a ! > such that for all x with < a ! x < " , f ( x ) ! L < " . Properties of Limits Theorem 1 . Let f ( x ) be a nondecreasing function on the interval a , b ( ) . Then for any c ! a , b ( ) , if L ! = lub f ( x ) a < x < c { } , then x ! c " lim f ( x ) = L " and if L + = glb f ( x ) c < x < b { } , then x ! c + lim f ( x ) = L + . Theorem 2 . Let a n { } n = 1 ! be a nondecreasing sequence. Then n !" lim a n = L where L = lub a n { } n = 1 ! . Theorem 3 . Let f ( x ) be a nonincreasing function on the interval a , b ( ) . Then for any c ! a , b ( ) , if L !...
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.

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Limits - Limits and Continuity Definitions of Limits x ! a...

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