44 2.2 Limit Theorems If 0<b<1, let c = 1 b and define xn = n √c = 1 an . Since c >1, it has been shown that limn→∞xn =1. This implies lim n→∞ an =lim n→∞ 1 xn =1. Exercises 2.2.1 Find the following limits: (a) lim n→∞ 3n2 −6n+7 4n2 −3 , (b) lim n→∞ 1+3n−n3 3n3 −2n2 +1 . 2.2.2 Find the following limits: (a) lim n→∞ √3n+1 √n+√3 , (b) lim n→∞ n r2n+1 n . 2.2.3 ▶Find the following limits if they exist: (a) lim n→∞ (pn2 +n−n), (b) lim n→∞ ( 3 pn3 +3n2 −n), (c) lim n→∞ ( 3 pn3 +3n2 −pn2 +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, limn→∞(b+br+br 2 +· · ·+brn). (b) lim n→∞ 2 10 + 2 102 +· · ·+ 2 10n . 2.2.5 Provide counterexamples for each of the following statements: (a) If {an}and{bn}are divergent sequences, then {an +bn}is a divergent sequence. (b) If {an}and{bn}are divergent sequences, then {anbn}is a divergent sequence. (c) If {an}and{an +bn}are divergent sequences, then {bn}is a divergent sequence.
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