limit $\lim_{x \to \infty}\frac{e^{e^x}}{e^{x^3}}$

Question

What is $$\lim_{x \to \infty}\frac{e^{e^x}}{e^{x^3}}$$

This is indeterminate form, so I can apply L'Hôpital's rule, but then limit becomes even harder.

Is there any smart manipulation?

Answer 1

Note that

$$\frac{e^{e^x}}{e^{x^3}}=e^{e^x-x^3}\to +\infty$$

indeed

$$e^x-x^3=e^x \left( 1-\frac{x^3}{e^x}\right)\to +\infty\cdot (1-0)= +\infty$$

Answer 2

$\lim_{x \to \infty}\frac{e^{e^x}}{e^{x^3}} = \lim_{x \to \infty}e^{e^x - x^3}$. Now you may probably use that $y:= e^x - x^3$ tends to infinity as $x$ tends to infinity, which gives $\lim_{y \to \infty}e^y = \infty$.