Proving a Limit Using the Definition of a Limit

[azatkgz]

Please,check my solution.

Homework Statement

Use definition of a Limit to prove it.
$$\lim_{n\rightarrow\infty}\frac{2n^2-\sqrt{n}}{3n^2+5logn}=\frac{2}{3}$$

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

 

[lippyka]

Since, for n>N, we have \frac{\sqrt{n}}{8n^2}<\frac{1}{4\epsilon^{2/3}}<\epsilon Therefore, |\frac{2n^2-\sqrt{n}}{3n^2+5logn}-\frac{2}{3}|<\epsilonHence, \lim_{n\rightarrow\infty}\frac{2n^2-\sqrt{n}}{3n^2+5logn}=\frac{2}{3}Yes, your solution is correct.