Evaluate:$$\lim\_{x\rightarrow \infty}\frac{\ln \left( \sqrt[x]{a\_1^x+a\_2^x+...+a\_n^x}-a\_1 \right)}{x},\ a\_1\ge a\_2\ge ...\ge a\_n>0$$
Lop does not work. I have no idea about this limit.  

I can prove $\ln(a\_1^x+a\_2^x+...+a\_n^x)\sim x\ln a\_1$ but I got stuck here.