Limit of $\frac{\log2^{x}\log\left(\log2^{x}\right)\log\left(\log\left(\log2^{x}\right)\right)^{p}}{\left(\log x\right)^{p}}$

Question

How to determine the following limit:

$$\lim_{x \to +\infty}\frac{\log2^{x}\log\left(\log2^{x}\right)\log\left(\log\left(\log2^{x}\right)\right)^{p}}{\left(\log x\right)^{p}}$$

Trying L'Hôpital's rule takes much time and I'm not even sure if we need use that just for once, so what is really the best way to determining the value of the limit?

---

I tried Wolframalpha ,but that was not helpful.

Answer 1

The numerator can be rewritten as

$$x\cdot p\cdot\log 2 \cdot (\log x + \log 2) \cdot \log(\log x + \log 2)$$

For $x > e^3$ we have that the numerator is greater than $ x \cdot (p\log 2)$

Thus the limit

$$L > \lim_{x\to\infty} p\log 2\cdot\frac{x}{(\log x)^p} \to \infty$$

since exponentials dominate any polynomial growth.