I am having trouble finding the largest lower bound (inf?) on this sequence given my epsilon.
$\frac{n^2 + 2n + 1}{2n^2 - n + 2} \to \frac{1}{2}$
So $\left|\frac{n^2 + 2n + 1}{2n^2 - n + 2}- \frac{1}{2}\right| = \left|\frac{5n}{4n^2 - 2n + 4}\right| \leq \frac{5}{2(n - 1)}$
In the last fraction, I simply did $4n^2 -2n + 4 > 4n^2 - 2n = 2(2n^2 - n)$
Let's say I choose epsilon to be $10^{-6}$ (good enough I think)
My problem is that for this rational function, I could probably make an upper bound like $\frac{1}{n-1}$ and it would still work, but i am not sure how to keep doing this to get a "largest" lower bound
EDIT: I should probably almost add a remark that I would run into a similar problem for other rationals too