Tried L'Hospital's rule, using logarithms and variable substitution. Guess I'm missing something or making mistakes. Thanks.
2026-05-15 15:34:47.1778859287
Any ideas on how to solve this? $\lim_{x\to \infty}x(\frac{1}{e}-(\frac{x}{x+1})^x)$
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Observe that, for $x\to \infty$, $$ \begin{split} \left(\frac{x}{x+1}\right)^x & = \exp\left[ -x \log\left(1+\frac{1}{x}\right)\right] \\ &= \exp\left[-x\left(\frac{1}{x}-\frac{1}{2x^2} + o(1/x^2)\right)\right] \\ & = \frac{1}{e}\exp\left(\frac{1}{2x} + o(1/x)\right) \\ & = \frac{1}{e} + \frac{1}{2e x} + o(1/x). \end{split} $$