Evaluate the limit $\lim_{x\to\infty}\frac{x^{\log_2 x}}{(\log_2 x)^x}$

Question

Evaluate the limit
$$\lim_{x\to\infty}\frac{x^{\log_2 x}}{(\log_2 x)^x}$$
I tried to evaluate this limit by taking natural log and using L'Hospital rule. However, it seems that both methods make the expression even harder to evaluate. I would be thankful for any help.

Answer 1

We have that

$$\frac{x^{\log_2 x}}{(\log_2 x)^x}=2^{(\log_2 x)^2-x\log_2(\log_2 x)}$$

and

$$(\log_2 x)^2-x\log_2(\log_2 x)=x\left(\frac{(\log_2 x)^2}x-\log_2(\log_2 x)\right)\to -\infty$$