I need to find
$\lim_{x\rightarrow0}f(x)$ for the following function:
$f:(0,+\infty)$
$f(x)=[1+\ln(1+x)+\ln(1+2x)+\dots+\ln(1+nx)]^\frac{1}{x}$
I tried writing the logarithms as products:
$\lim_{x\rightarrow0}[1+\ln(1+x)(1+2x)\dots(1+nx)]^\frac{1}{x}$
and as a sum and nothing is getting me anywhere.
Also I know I have to use the formula: $\lim_{x\rightarrow0}(1+x)^\frac{1}{x}=e$
Can someone please help me?
Thank you very much!