Is this a valid step in a convergence proof? I'm asked to say what the following limit is, and then prove it using the definition of convergence. 
$\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$. 
Is it valid to say that the sequence behaves like $\dfrac{3n^2}{4n^2}$ for large n?

Just to expand a bit on the above solution, just to the left of the first (*) we need $\frac{5}{16n}< ϵ $ which implies that we need $\frac{5}{16ϵ} < n$ in case it wasn't already plainly obvious.  Then setting N > $\frac{5}{16ϵ}$ guarantees the result for all n≥N.      
