Help in evaluating $\lim_{x \rightarrow \infty} \frac{1000^x}{x^x} = 0$ I suspect that
$$\lim_{x \to \infty} \frac{1000^x}{x^x} = 0.$$
However, I do not know how to prove that this is the case. Any help would be greatly appreciated.

Write $$\frac{1000^x}{x^x} = \exp(x (\ln 1000 - \ln x))$$
What can you say about $\ln 1000 - \ln x$, and then about $x (\ln 1000 - \ln x)$, as $x \to +\infty$?
