Limit $\lim\limits_{n\to\infty} \sqrt[n]{\frac1{\sqrt3}\left(\left(\frac{1+\sqrt3}2\right)^n-\left(\frac{1-\sqrt3}2\right)^n\right)}$

For the beginning prove the following facts $$ \lim\limits_{n\to\infty} \sqrt[n]{a}=1 \quad\text{ for }\quad a>0 $$ $$ \lim\limits_{n\to\infty} \sqrt[n]{x^n-y^n}=x\quad\text{ for }\quad x>|y| $$ then apply them to the limit $$ \lim\limits_{n\to\infty}\sqrt[n]{a(x^n-y^n)} $$