Proving limits without limit theorems I have to prove the following limit without using any limit theorems. I can only do so by using the Archimedean Proprety and the definition of a limit.
I have to prove:
$$\lim_{n\to\infty} \frac{n^2 - 2}{3n^2 + 1} = \frac13$$

I'll help get you started. You want to show that N>n => |(n^2 - 2)/(3n^2 + 1) -1/3| < epsilon. (Definition of convergence of a sequence). 
First, get a common denominator. 
|-7/(9n^2+3)|< epsilon
Then you're gonna want to solve for n in terms of epsilon and set N equal to the value you get for n. 
