$$\lim _{x\to \infty }\left(\frac{2^{\frac{1}{10}\left(log_2x\right)^{\frac{1}{2}}}}{log_2x}\right)$$
Apply substitution $x\mapsto x^4$ and use L'Hôpital's Rule:
$$\lim _{x\to \infty }\frac{\frac{1}{10} 2^{\frac{x}{10}}\ln 2}{2x}$$ Especially after $x>10$, it is clear that the numerator is much greater than the denominator. Thus, the limit diverges to infinity. In fact, we can make the denominator look even more embarrassingly tiny by continuing to make substitutions: $$=\frac{\ln 2}{20}\lim _{x\to \infty }\frac{2^{\frac{x}{10}}}{x}$$
$$=\frac{\ln 2}{200}\lim _{x\to \infty }\frac{2^x}{x}$$
Thus, we have
$$\lim _{x\to \infty }\left(\frac{2^{\frac{1}{10}\left(log_2x\right)^{\frac{1}{2}}}}{log_2x}\right)\color{red}{\text{ diverges to positive infinity}}$$