E-N Definition of Limit with a minus in the denominator

I'm working on a problem, using E-N limits of which I have not seen an example of before, or could find an example of on the internet.

$$s_n = \frac{4n^2 + \sin(n)}{2n^2-1}$$

By inspection, $\lim\limits_{n\to\infty}s_n=2$.

$$s_n - 2 = \frac{4n^2 + \sin(n)}{2n^2-1} -2 = \frac{\sin(n) + 2}{2n^2 - 1}$$

The solution to this problem is not important, all I would like to know is how to deal with the $-1$ in the denominator, as every other problem has a $+1$ in and can be easily manipulated

Many thanks!

Solution 1:

Choose $N = \sqrt{\frac{3/\epsilon + 1}{2}}$.

Then $\forall n\in\mathbb{N}$ such that $n > N$,

$|\frac{\sin(n)+2}{2n^2-1}| \le |\frac{3}{2n^2-1}| < |\frac{3}{2(\sqrt{\frac{3/\epsilon + 1}{2}})^2-1}| = |\frac{3}{2(\frac{3/\epsilon + 1}{2})-1}| = |\frac{3}{3/\epsilon + 1-1}| = |\epsilon| = \epsilon$.