I'm trying to solve $$\lim\limits_{x \to \infty}\frac{\ln^{1000} x}{x^5}$$ Here's what I get: $$e^{\lim\limits_{x \to \infty}\ln{\frac{\ln^{1000}x}{x^5} }}$$ Dropping the $e$ for ease, $$\lim\limits_{x \to \infty} 1000\ln{(\ln{(x)})} - 5 \ln{x} $$
Now I have $\infty - \infty$.. I know there must be a next step, but I don't know what it would be.