QUIZ 2. ADVANCED CALCULUS 1, FALL 2020 1. (5 pts) Prove using the definition of the limit (find N (ε) for every ε) that 2n2 + 1 2 lim = . 2 n→∞ 3n − 1 3 2. (5 pts) Which of the following statements imply that the sequence {xn } is bounded? Prove your answer. (A) There exist ε > 0 and a ∈ R so that for every n ∈ N satisfying n > 100 we have |xn − a| < ε. ε (B) There exists a ∈ R such that for every ε > 0 we have |xn − a| < ε for all n ∈ N satisfying n < 100 . ε (C) There exist ε > 0 and a ∈ R so that we have |xn − a| < ε for . every n ∈ N satisfying n < 100 ε (D) There exists ε > 0 such that for every a ∈ (−∞, −10) ∪ (10, ∞) we have |xn − a| > ε for every n ∈ N satisfying n > 100 . ε 3. (5 pts) Suppose that two sequences {xn } and {yn } converge to the same limit a ∈ R. Define a sequence {zn } by z2n = xn , z2n−1 = −yn for all n ∈ N. For which numbers a does the sequence {zn } converge? Prove your answer. 1
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