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1. (5 pts) Prove using the definition of the limit (findN(?) for every?) thatlimn??2n2+ 13n2?1=23.

2. (5 pts) Which of the following statements imply that the sequence{xn}is bounded? Prove your answer.

(A) There exist? >0 anda?Rso that for everyn?Nsatisfyingn >100?we have|xn?a|< ?.
(B) There existsa?Rsuch that for every? >0 we have|xn?a|< ?for alln?Nsatisfyingn <100?.
(C) There exist? >0 anda?Rso that we have|xn?a|< ?foreveryn?Nsatisfyingn <100?.
(D) There exists? >0 such that for everya?(??,?10)?(10,?)we have|xn?a|> ?for everyn?Nsatisfyingn >100?.

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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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Tags:

mathematics

calculus

Advanced calculus

conditions

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