Seq_hnd - n n b →∞ both exist and are finite. Then (i)...

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Sequences Theorems on Sequences: I. Modeling Theorem Let { } a n be a sequence, let f n a n ( ) = , and suppose that f x ( ) exists for every real number x 1. (i) lim ( ) x f x L →∞ = , then lim ( ) n f n L →∞ = . (ii) lim ( ) x f x →∞ = ∞ - ∞ (or ) , then lim ( ) n f n →∞ = ∞ - ∞ (or ) . II. Let r be a real number. Then (i) lim n n r →∞ = 0 if r < 1 (ii) lim n n r →∞ = ∞ if r 1 III. Sandwich Theorem If { } a n , { } b n , and { } c n are sequences and a b c n n n for every n and if lim lim n n n n a L c →∞ →∞ = = , then lim n n b L →∞ = . IV. Let { } a n be a sequence. If lim n n a →∞ = 0 , then lim n n a →∞ = 0 V. Suppose that lim n n a →∞ and lim
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Unformatted text preview: n n b →∞ both exist and are finite. Then (i) lim lim n n n n ca c a →∞ →∞ = for any real number c (ii) lim( ) lim lim n n n n n n n a b a b →∞ →∞ →∞ + = + (iii) ( 29 ( 29 lim lim lim n n n n n n n a b a b →∞ →∞ →∞ = (iv) If lim n n b →∞ ≠ , then lim lim lim n n n n n n n a b a b →∞ →∞ →∞ = . VI. A bounded, monotonic (nondecreasing or nonincreasing) sequence has a limit....
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This note was uploaded on 03/09/2008 for the course CALC 1,2,3 taught by Professor Varies during the Spring '08 term at Lehigh University .

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