How to convert this limit using the squeeze theorem ?
$$\lim_{n\to ∞}\sqrt[n]{3^n+(2|\sin(n^n)|)^n}$$
Note that
$$0 \le |\sin(n^n)| \le 1 $$
So, we can write
$$\color{red}{\sqrt[n]{3^n} \le \sqrt[n]{3^n+(2|\sin(n^n)|)^n} \le \sqrt[n]{3^n+2^n}} \le \color{blue}{\sqrt[n]{2\cdot 3^n}}$$
I can roughly imagine how we got $\color{red}{\mathit{the\ three\ parts}}$ of this expression, but I don’t understand how to get the $\color{blue}{\mathit{fourth~part}}$ of this expression and what to do next. Can you please show me what steps i should take next.