Prove using the definition of the limit that limn→∞1+n1−2n=−12 Prove using the definition of the...

Jay Mckay

Jay Mckay

Answered

2022-01-24

Prove using the definition of the limit that limn1+n12n=12
Prove using the definition of the limit that limnn2+12n1=

Answer & Explanation

Brenton Pennington

Brenton Pennington

Expert

2022-01-25Added 15 answers

I solved only the first one
Note that |1+n12n+1n|
=|2+2n+12n2(12n)|
=|32×112n|
For every ξ=0 given we can choose n such that
|32×112|=32(2N1)<ξ
2N1>32ξ
N>(1+32ξ)12
N>(12+34ξ)
Hence, nN
|1+2n12n(12)|<ξ
Thus, limn|1+2n12n|=12
(From def of limit)

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