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.

Hint: write  ${1000^x\over x^x}= ( {1000\over x})^x$. Note that $0<{1000\over x} $ and that eventually, ${1000\over x}<{1\over 2}$. Then compare (use the squeeze theorem) with ${1\over 2^x}$.
