Calculate the following limits:

1.) $\displaystyle \lim_{x \to \infty} \frac{\ln{x}}{x^{\frac{2}{3}}}$

I know the answer is 0. Can I simply use L'Hopitals?

So, $\displaystyle \frac{\frac{1}{x}}{\frac{2}{3}x^{\frac{-1}{3}}}$? Something like that?

And then,

2.) Does $\displaystyle \int_{1}^{\infty} \frac{e^{-x}}{\sqrt{x}}\, dx$ converge? Justify your answer.

No idea how to justify, but my calculator gives .2788 as an answer so it def does converge.