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> Evaluate $$\lim\_{n\to \infty} \frac{\ln(n)^n}{5n^n}$$
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$$\lim\_{n\to \infty} \frac{\ln(n)^n}{5n^n}=\lim\_{n\to \infty} \frac{n\cdot \ln(n)}{5^nn^n}=\lim\_{n\to \infty} \frac{\ln(n)}{5^nn^{n-1}}$$

$\ln(n)<<5^nn^{n-1}$ and therefore $$\lim\_{n\to \infty} \frac{\ln(n)^n}{5n^n}=0$$

Is there a way to solve it without L'Hospital? and $\ln(n)<<5^nn^{n-1}$?