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Unformatted text preview: 5. Consider the Function f ( x ) = 1 1 + x 2 , x ∈ R . Prove, using the defnition, that (a) lim x →±∞ f ( x ) = 0 . (1) (b) f is uniFormly continuous. 6. Given the convergent sequences { a n } n ∈ N ⊂ R and { b n } n ∈ N ⊂ R , let a ∈ R and b ∈ R be the corresponding limits, i.e., lim n →∞ a n = a ; lim n →∞ b n = b. Prove, using the defnition oF limit, the Following statements: (a) lim n →∞ ( a n + b n ) = a + b . (b) lim n →∞ a n b n = ab . (c) IF b 6 = 0, lim n →∞ a n b n = a b . 7. Prove the Following statement: A Function f : I → R is continuous at x ∈ I iF and only iF ∀{ x n } n ∈ N ⊂ I such that x n → x , it Follows that f ( x n ) → f ( x )....
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 Fall '09
 Math, Calculus, lim, Continuous function, Limit of a sequence, Uniform continuity, Prove

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