So I'm completely lost on this so far:
The question reads use the epsilon definition to prove the following:
$\lim_{n\to\infty}(n^2+n+1)/(2n^2+3) = 1/2$
so we have to show that for any $\epsilon>0$, there is an $M>0$ such that $n>M \Rightarrow |a_n-L|<\epsilon$
normally when I solve these I set an $\epsilon>0$, and then try to find an M(that depends on ε) such that n>M satisfies the definition.. anyway this is what I tried
\begin{align*} \left |\frac{n^2 + n + 1}{2n^2+3} - \frac{1}{2}\right| < \epsilon &\Rightarrow \left|\frac{n^2+n+1}{2n^2+3}-\frac{\frac{1}{2}(2n^2+3)}{2n^2+3}\right| < \epsilon \\ &\Rightarrow \left|\frac{n-\frac{1}{2}}{2n^2+3}\right| < \epsilon \\ \end{align*}
now at this point everything I've done has led nowhere .. I can't seem to isolate the n and so I'm hoping there is maybe some trick I've missed or something..
I don't necessarily want a full rigorous answer, just a thought on what I might try from there (or maybe a different strategy altogether) ... Thanks