Sequence limit help

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Hello all

For the sequence (n^2 + n -1 )/ (3n^2 +1) I am trying to find

lim (n^2 + n -1)/ (3n^2 +1) = 1/3
n -> 00


My solution"

= 1 - (2n^2 - n +2/ 3n^2 +1) = 1- r(sub-n)

We need to prove that r(sub-n) approaches 2/3.

Hence 2n^2 - n + 2 < 2n^3
3n^2 + 1 > 3n^2

The remainder r sub n is 2n/3. Is this right? How would I find the remainder?

Also how do I find an N such that n > N and the difference between f(x) and L is less than 1/ 10, 1/100, and 1/1000.

Any help would be appreciated.

Thanks