L'Hopital's Rule/Limit troubles

[tangents]

Hello,
Well my math class finally started integrals after finishing H's Rule. But while I was doing my HW, I wouldn't get the correct answer but was sure I was following the rule. The rule is simple saying that if taking the limit gives indeteminate form ,then find dervative and then take limit. So anyway here are the two questions that I can't get
1) lim x->0+ (Ln[sin[x]])/(Ln[sin[2x]])
I took dervative and got Cot[x]/2cot[2x]. but since sin 0=0 i continued to get un defined so i took dervative again but got sin in the answer so i knew i was having some problem. The answer should be one but i can't seem to get it

2) Lim x->0 (xcos[x]+e^-x)/x^2
i did the dervative twice until the denominater was a constant and got the answer 0 but that isn't in domain. the correct answer is infinite but I again don't know how the textbook got that

Any ideas why I can't get the right answer =/

 

[arildno]

As for 2), check the limit of your original numerator as x goes to 0.

As for 1), understand why we have:
$$\frac{\cot(x)}{2\cot(2x)}=\frac{\sin(2x)}{2\sin(x)}\frac{\cos{x}}{\cos(2x)}=\frac{\cos^{2}(x)}{\cos(2x)}$$