Specifically, I ran into trouble when trying to solve $$\lim\limits_{n \to \infty} \int_1^n x^{-n} \, dx.$$
The integral is very easy to solve but what next? The limit $\displaystyle\lim\limits_{n\to \infty} \frac{n^{-n+1}-1}{-n+1}$ looks like $\displaystyle\lim\limits \frac{\infty^{-\infty}}{-\infty}$ which I'm not sure how to solve. Can I apply L'Hopital Rule here?