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Finding the limit of a sum for a specific case

Find the limit of this sum:
$$\lim_{n\to\infty}\frac{\sqrt{2^n+n^2}+n^2}{n^2+2n^4+2^n+n^2\sqrt{2^n+n^2}}$$

I want to try and use Squeeze theorem, but I’m not quite sure if this is the way to go. It would be great if I got some tips!

A:

We have
$$n^2+2n^4+2^n+n^2\sqrt{2^n+n^2}\ge n^2+2n^4+2^n=\frac{n^4+2n^4+2^n}2+n^2=\frac{(n+2)^2}2+n^2$$
hence, since $\displaystyle\lim_{n\to\infty}\frac{(n+2)^2}{2n^2}=1$, the limit is $$1+\lim_{n\to\infty}\frac{n^2+n^2\sqrt{2^n+n^2}}{n^2+2n^4+2^n}=1+\lim_{n\to\infty}\frac{n^2+n

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