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How do you evaluate the limit

$$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$$

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You could try to evaluate the limit of the logarithm of this. –  Did Apr 25 '11 at 7:46
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up vote 7 down vote accepted

I think we should be witty about how we write it. How about we consider instead the limit $$ \lim_{x \to \infty} \frac{x^x}{(x+1)^x (x+1)} = \lim_{x \to \infty} \left ( \frac{x}{x+1} \right )^{x} * \frac{1}{x+1} $$

I think that this is suggestive of a proof?

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How about using squeeze theorem? Try squeezing this as $0 \leq \frac{x^x}{(x+1)^{x+1}} \leq \frac{x^x}{x^{x+1}} = \frac{1}{x}$.

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+1 Very nice! I don't think you can get a simpler solution than this one. –  Adrián Barquero Apr 26 '11 at 4:24
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