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The limit is $$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$$

Experimentally, this limit appears to converge to ${1 \over e}$, but I can't figure out how to solve it.

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    Mathematica confirms the limit is 1/e; as for figuring out how this result was arrived at, just to give you a hint, this is an ∞-∞ type indeterminate form. Manipulate it into something where L'Hôpital can apply (you may also have to invoke logarithmic differentiation at some point). Good luck!2010-08-14
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    I did manipulate it and use L'Hôpital's rule, but then it seemed to explode into further complexity, so I gave up. If this is the proper way to solve the limit, then I'll try to muck through all the details.2010-08-14
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    You will have to take logarithms at some point and come up with an expression whose limit is -1. But since this is the limit for the logarithm, the limit for the original expression should be $\exp(-1)$.2010-08-14

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