Prove using the definition on the limit that \lim_{n\to\infty}\frac{1+n}{1-2n}=\frac{-1}{2}

slaggingV 2021-08-19 Answered
Prove using the definition on the limit that
limn1+n12n=12
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Expert Answer

wheezym
Answered 2021-08-20 Author has 103 answers

|1+n12n+12|
=|2+2n+12n2(12n)|
=|32112n|
For every ϵ>0 given we can choose N that
|32112=32(2N1)|<ϵ

2N1>32ϵ
N>(1+32ϵ)12
N>(12+34ϵ)
Hence nN
|1+2n12n(12)|<ϵ
Hence
limn|1+2n12n|=12

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