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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.2011-04-25

2 Answers 2

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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.2011-04-26