<s>[INST] Finding $\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}}$ 
Find the limit: $\displaystyle\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}}$

My attempt: 
$\begin {align}\displaystyle\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}} &=
\lim _{n\to \infty}\frac{\sqrt{n+n^4\sqrt{1+\frac{1}{n^2}}}}{n^2\sqrt{3+\frac{1}{n^2}}}\\ &=
\lim _{n\to \infty}\frac{n^2\sqrt{1+n^3\sqrt{1+\frac{1}{n^2}}}}{n^2\sqrt{3+\frac{1}{n^2}}}\\&= 
\lim_{n\to \infty}\frac{\sqrt{1+n^3\sqrt{1+\frac{1}{n^2}}}}{\sqrt{3+\frac{1}{n^2}}}\end{align}$
Which looks like is equal to $\infty$ because of the $n^3\cdot 1$ but it's wrong. 
What am I doing wrong here?
 [/INST] Here are the steps
$$\lim\limits _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}}
$$
$$= \lim\limits _{n\to \infty}\sqrt{\frac{n+\sqrt{n^2+1}}{3n+1}} $$
$$= \sqrt{ \lim\limits _{n\to \infty} \frac{n+\sqrt{n^2+1}}{3n+1}} $$
$$= \sqrt{ \lim\limits _{n\to \infty} \frac{n+\sqrt{n^2\left(1+\frac{1}{n^2}\right)}}{3n+1}} $$
$$= \sqrt{ \lim\limits _{n\to \infty} \frac{n+|n|\sqrt{1+\frac{1}{n^2}}}{3n+1}} $$
$$= \sqrt{ \lim\limits _{n\to \infty} \frac{1+\sqrt{1+\frac{1}{n^2}}}{3+\frac{1}{n}}} $$
$$= \sqrt{ \frac{1+\sqrt{1+0}}{3+0}} $$
$$= \sqrt{ \frac{2}{3}} $$
</s>