# Definition of a Limit.

1. Oct 13, 2007

### azatkgz

1. The problem statement, all variables and given/known data
Use definition of a Limit to prove it.
$$\lim_{n\rightarrow\infty}\frac{2n^2-\sqrt{n}}{3n^2+5logn}=\frac{2}{3}$$

3. The attempt at a solution

for $$\forall\epsilon>0$$ $$\exists N$$ such that $$\forall n>N$$
we have
$$|\frac{2n^2-\sqrt{n}}{3n^2+5logn}-\frac{2}{3}|<\epsilon$$

$$\frac{\sqrt{n}}{8n^2}<|\frac{-3\sqrt{n}-10logn}{3(3n^2+5n)}|<\epsilon$$

We can choose $$N=\frac{1}{4\epsilon^{2/3}}$$