$ \lim_{n \to \infty} (1+ \frac{3}{n^2+n^4})^n$ I've tried to solve the limit
$$     \lim_{n \to \infty} (1+ \frac{3}{n^2+n^4})^n$$
but I'm not sure.
$    (1+ \frac{3}{n^2+n^4})^n =   \sqrt [n^3]{(1+ \frac{3}{n^2+n^4})^{n^4}} \sim    \sqrt [n^3]{(1+ \frac{3}{n^4})^{n^4}} \sim   \sqrt [n^3]{e^3 }  \rightarrow 1$
Is it right?

Yes, I think it's right. Of course, you might have to justify that your approximations still preserve the limit. 
Another way would be to write 
$$
\left(1+ \frac{3}{n^2+n^4}\right)^n=\exp\left( n\log\left(1+ \frac{3}{n^2+n^4}\right)\right)\sim\exp\left( \frac{3n}{n^2+n^4}\right)\to e^0=1.
$$
