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Find $\lim\limits_{x\to \infty} x(\frac{1}{e} - (\frac{x}{x + 1})^x)$.

I've tried several methods, like $e$ to the power $ln$ of expression or to make a substitution. The possible way to find the answer, I think, is to make a substitution, which I can't see now.

Nikita Tkachuk
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    See https://math.stackexchange.com/questions/73243/limit-of-x-left-left1-frac1x-rightx-e-right-when-x-to-infty – Martin R Nov 01 '22 at 16:01
  • Consider $f(h)=(1+h)^{-h}$ for $h\neq0$ and $f(0)=1/e$. $f$ is continuous in $[0,1]$ and differentiable in $(0,1]$ for example. Your limit can be expressed as $$\lim_{h\rightarrow0+}\frac{f(h)-f(0)}{h}$$ – Mittens Nov 01 '22 at 16:03
  • @Andrei, Yes, thank you very much. The simple substitution x = 1/t does its job. – Nikita Tkachuk Nov 02 '22 at 15:03

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