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# Limits - Limits and Continuity Definitions of Limits For...

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Limits and Continuity Definitions of Limits x ! a lim f ( x ) = L . For all ! > 0 there is a ! > 0 such that for all x with x ! a < " and with x ! a , f ( x ) ! L < " . n ! " lim a n = L For all ! > 0 there is an N such that for all n ! N , a n ! L < " . x ! a + lim f ( x ) = L For all ! > 0 there is a ! > 0 such that for all x with 0 < x ! a < " , f ( x ) ! L < " . x ! a " lim f ( x ) = L For all ! > 0 there is a ! > 0 such that for all x with 0 < 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 ! = glb f ( x ) a < x < c { } , then x ! c " lim f ( x ) = L " and if L + = lub f ( x ) c < x < b { } , then x ! c + lim f ( x ) = L + . Theorem 4 . Let a n { } n = 1 ! be a nonincreasing sequence. Then n ! " lim a n = L where L = glb a n { } n = 1 ! . Theorem 5 . (Squeeze) Suppose that lim x ! a f ( x ) = L and lim x ! a g ( x ) = L . Suppose also that for all x ! a , f ( x ) ! h ( x ) ! g ( x ) . Then lim x ! a h ( x ) = L .

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