L'hospital's Rules tricky proofs need help

Edit [Moderator]: Moved from General to Science & Mathematics.

I am introducing myself, good lookin female -- double major for BS in physics and BA in technical theatre with math minor -- freaking out over calculus ineptitude and jealous of the only 2 people in my class who actually "get it". check out my webpage if you want to get to know me Edit (Moderator): Link removed I Just need some reasurance or correction for two problems that use L'hospital's Rule.

1. The lim as x->infinity of (1 + a/x) to the power bx.

I managed to get lim of (a(bsquared))/ (1 +a/x) = + infinity with some implicit differentiation and I could be totally wrong but at some point I need to get the right answer into the form e but I am having a brain cramp or something and I can't find it anywhere I was thinking the answer was e to the x power but I have a feeling it can't be that easy and there are more terms floating around that need to be included in the e form.


2. So when proving this: The lim as x-> infinity for (lnx)/(x to the power p) = 0

It is because when we use L'hospital's rule we get eventually lim (1/x)/(P!)
Right? but where do I go from here to get to prove that for any p>0 that the logarithmic function approaches infinity more slowly than any power of x. Do I need to do L'hospital's again and if I do what is the logarithmic form I'm supposed to sub in to get back into a form where the lim of the function is determinate one and is zero?