On certain limits related to the number \(e\) (Q2711277)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On certain limits related to the number \(e\) |
scientific article |
Statements
6 August 2002
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functional inequalities.
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limits
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mean values
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differentiable functions
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increasing functions
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concave functions
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On certain limits related to the number \(e\) (English)
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The author points out that the limit results offered, such as NEWLINE\[NEWLINE\lim_{x\to\infty}[x^{-x}(x+1)^{x+1}-x^x (x-1)^{1-x}]=e=\lim_{x\to\infty}[(x+1)^{-x}(x+2)^{x+1}-(x+1)^x x^{1-x}],NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\lim_{x\to\infty}[(x+1)^{1-x}(x+2)^{x+1}-(x+1)^{x+2}x^{-x}]=e/2,NEWLINE\]NEWLINE are either known or easy consequences of the Bernoulli-l'Hospital rule. But he looks also inside the square brackets, e.g., offering a proof that the functions \(x\mapsto [(x+1)/x]^x\) and \(x\mapsto(x+1)^{x+1}/x^x\) are increasing and concave (strictly) for \(x>0.\)
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