> 
> Evaluate
>  $$I=\lim\_{x\to 0}\dfrac{(1+x)^{\frac{1}{x}}-(1+2x)^{\frac{1}{2x}}}{x}$$
> 
> 
> 

My try: Use L'Hôpital's rule. We have 
$$I=\lim\_{x\to0}(1+x)^{1/x}\left(\dfrac{x}{x+1}-\ln{(x+1)}\right)-(1+2x)^{1/2x}\left(\dfrac{4x}{1+2x}-2\ln{(1+2x)}\right)$$

I'm stuck. Thank you very much for you help