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using the Epsilon-N definition of the limit prove this !

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using the Epsilon-N definition of the limit prove this !
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Answered Same Day Dec 20, 2021

Solution

Robert answered on Dec 20 2021
122 Votes
So let � > 0. We have to show that ∣∣∣∣n · (√n2 + 2− n)− 1∣∣∣∣ <�∣∣∣∣ 2n√n2 + 2+ n − 1
∣∣∣∣ < �∣∣∣∣n−
√
n2 + 2
n+
√
n2 + 2
∣∣∣∣ < �
1−
√
1+ 2n2
1+
√
1+ 2n2
∣∣∣∣ < �
Now for n > n0 we have
∣∣ 2
n2
∣∣ < �. Let t0 = √1+ �. Therefore we have
1− t0
1+ t0
which is an a
itrary...
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