A lot of terms to calculate lim I'm trying to prepare for exam and I came across a limit to calculate. 
$$
\lim_{n->\infty} \frac{2^n + \left(1+\dfrac{1}{n^2}\right)^{n^3} + \dfrac {4^n}{n^4}}{\dfrac {4^n}{n^4} + n^3\cdot 3^n}
$$
When I'm trying to extract $4^n$ I end up with nothing. 
And I managed to tell that $(1+\frac{1}{n^2})^{n^3}$ goes to infinity, bounding it with $2$ and $3$. Cause I know that $(1+\frac{1}{n^2})^{n^2}$ goes to $e$ as $n$ is increasing. 
I'd appreciate some help or tips on that one, cause I'm stuck with it for over an hour and I couldn't find any clues how to solve it on-line.
Chris

A bit (very) fast and loose but multiply by one AKA $\displaystyle \frac{n^4}{n^4}$ to get 
$$
\begin{align}\frac{n^4(2^n+e^n)+4^n}{4^n+n^73^n}&=\frac{n^4(2^n+e^n)}{4^n+n^73^n}+\frac{4^n}{4^n+n^73^n}\\&=\frac{n^4(2^n+e^n)}{4^n+n^73^n}+\frac{1}{1+n^7\cdot(\frac34)^n}
\\&\approx\frac{n^4e^n}{4^n}+1
\\&\approx 1.
\end{align}$$
for $n$ large.
i.e. $4^n$ is the Daddy here.
