How would one calculate $\lim_{n\to\infty}n((1+\frac{1}{n})^n-e)$? What I have thought about this is:


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*we may use L'Hopstal's rule to calculate $\lim_{n\to\infty}\frac{((1+\frac{1}{n})^n-e)}{\frac{1}{n}}$, both the numerator and denominator goes to 0 as n goes to infinity. But calculating the derivative of $(1+1/n)^n$ seems to be very complicated.

*Using Taylor series to calculate the dominant terms of $(1+1/n)^n$, but I'm not really sure if it makes sense to let $"n=\infty"$. Equivalently maybe we can expand $(1+x)^{1/x}$ at $x=0$, but it's not defined. 
Maybe I wasn't on the right track. Even if the solution uses a different approach, I would still love to know how to expand $(1+x)^{1/x}$. Thanks for any suggestions. 

hint
$$f (x)=(1+x)^\frac 1x=e^{\frac {1}{x}\ln (1+x)} $$
$$\ln (1+x)=x-\frac {x^2}{2}(1+\epsilon (x)) $$
$$f (x)=e.e^{-\frac {x}{2}(1+\epsilon (x))} $$
$$=e\Bigl (1-\frac {x}{2}(1+\epsilon (x)\Bigr)$$
The limit will be $$-\frac {e}{2} $$
