Introduction to Mathematical Analysis I - Second Edition

40 2.2 LIMIT THEOREMS For each ε > 0, choose N > b − 1 ε . It follows that for n ≥ N , | a n − 1 | = a n − 1 < b − 1 n ≤ b − 1 N < ε . Thus, lim n → ∞ a n = 1. In the case where b = 1, it is obvious that a n = 1 for all n and, hence, lim n → ∞ a n = 1. If 0 < b < 1, let c = 1 b and define x n = n √ c = 1 a n . Since c > 1, it has been shown that lim n → ∞ x n = 1. This implies lim n → ∞ a n = lim n → ∞ 1 x n = 1 . Exercises 2.2.1 Find the following limits: (a) lim n → ∞ 3 n 2 − 6 n + 7 4 n 2 − 3 , (b) lim n → ∞ 1 + 3 n − n 3 3 n 3 − 2 n 2 + 1 . 2.2.2 Find the following limits: (a) lim n → ∞ √ 3 n + 1 √ n + √ 3 , (b) lim n → ∞ n r 2 n + 1 n . 2.2.3 I Find the following limits if they exist: (a) lim n → ∞ ( √ n 2 + n − n ) . (b) lim n → ∞ ( 3 √ n 3 + 3 n 2 − n ) . (c) lim n → ∞ ( 3 √ n 3 + 3 n 2 − √ n 2 + n ) . (d) lim n → ∞ ( √ n + 1 − √ n ) . (e) lim n → ∞ ( √ n + 1 − √ n ) / n . 2.2.4 Find the following limits. (a) For | r | < 1 and b ∈ R , lim n → ∞ ( b + br + br 2 + · · · + br n ) . (b) lim n → ∞ 2 10 + 2 10 2 + · · · + 2 10 n . 2.2.5 Prove or disprove the following statements: