In other words, do you mean \(\displaystyle \lim_{x\to \infty} x^2+ 2x- \frac{6}{3x^2}+ 2\), which is what you wrote, or do you mean \(\displaystyle \lim_{x\to\infty}\frac{x^2+ 2x- 6}{3x^2+ 2}\), which is more likely?
For x very very large, the highest power of x "dominates". For example, with x= 100, \(\displaystyle 3x^2+ 2= 30000+ 2= 30002\) and if x= 1000, \(\displaystyle x^2+ 2= 3000000+ 2= 3000002\).