Need help proving a limit using the definition of a limit.

Question

I know the definition of the limit and how it applies in this case, but I don't know how to reach the end of the proof. The limit is:

$\lim_{n\rightarrow \infty}{\dfrac{2n^2-3n+1}{n^2-n+7}}=2$

I got to:

$\dfrac{n+13}{n^2-n+7}<\epsilon$

Thank you.

Answer 1

Use that $$\frac{n+13}{n^2-n+7}<\frac{13}{n}$$ since $n>0$ we get by cross multiplication $$n^2+13n<13(n^2-n+7)$$ and this is equivalent to $$0<12n^2-26n+91$$

Answer 2

You could use L'Hopital''s Rule, i.e. derive both the nominator and the denominator of your second expression to see that it converges to 0. According to L'Hopital, the same holds for the original expression!