$\lim _{x\to \infty }\left(\sqrt[x]{3^x+\left(2abs\left(sin\left(x^x\right)\right)\right)^x}\right)$

Question

So my calc teacher just dropped this limit on us in the second week. We're supposed to solve it using the Squeeze Theorem, but I have absolutely no clue what to do. $$\lim _{x\to \infty }\left(\sqrt[x]{3^x+\left(2abs\left(sin\left(x^x\right)\right)\right)^x}\right)$$

Answer 1

$$\sqrt[x]{3^x}\leq \sqrt[x]{3^x+(2|\sin(x^x)|)^x}\leq\sqrt[x]{3^x+2^x}$$

We have $$\lim_{x\to \infty}\sqrt[x]{3^x}=3$$ We also have $\lim_{x\to \infty}\sqrt[x]{3^x+2^x}=\lim_{x\to \infty}\sqrt[x]{3^x(1+(\frac{2}{3})^x})=\\3\lim_{x\to \infty}\sqrt[x]{(1+(\frac{2}{3})^x})=3$