i am stuck on finding a value for N, and how to simplify the equation in the absolute value
2026-04-14 11:27:56.1776166076
prove straight from the definition of convergence that (7n^2 -4)/(2n^2 + n +1) convergers to 7/2
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See my comment for future questions. Since you're new, here's a start.
With a few upper bounds, finding a suitable $N$ should not be too hard.
$$\begin{align}\left|\frac{7n^2-4}{2n^2+n+1}-\frac{7}{2}\right| & = \left|-\frac{7n+15}{2\left(2n^2+n+1\right)}\right| \\[5pt] & \le \left|\frac{\color{blue}{22n}}{2\left(2n^2\color{red}{+n+1}\right)}\right| \\[5pt] & \le \frac{22n}{2\left(2n^2\right)} \\[5pt] & = \frac{11}{2n} \end{align}$$ You can easily make $\tfrac{11}{2n}$ arbitrary small by taking $n$ sufficiently large.