## Description

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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## Explanation & Answer

Attached.

1

2n2 + 1 2

Note that n 1( 5n + 3 9n2 ) and 0 + 1 1 . We have lim 2

= by definition:

n →+ 3n − 1

3

2

2

2n2 + 1 2 3 ( 2n + 1) − 2 ( 3n − 1)

5

1

1

1

n N ( ) = + 1

− =

=

.

2

2

2

3n − 1 3

3 3n − 1

3 3n − 1 n N ( )

...