I have three advanced calculus question, need help
Question 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?.
3. (5 pts) Suppose that two sequences{xn}and{yn}converge to thesame limita?R.Define a sequence{zn}byz2n=xn, z2n?1=?ynfor alln?N.For which numbersadoes the sequence{zn}converge?Prove your answer.
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