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Is it possible to determine the limit

$$\lim_{x\to0}\frac{e^x-1-x}{x^2}$$

without using l'Hopital's rule nor any series expansion?

For example, suppose you are a student that has not studied derivative yet (and so not even Taylor formula and Taylor series).

  • 9
    Why don't you want to use them ?2012-08-18
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    @Belgi: take it as a challenge or a curiosity2012-08-18
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    There were couple of questions about limits with l'Hoptial's; you might want to check out [1](http://math.stackexchange.com/questions/182999/how-to-evaluate-lim-limits-x-rightarrow-inftyex-e-1-frac1x?rq=1) and [2](http://math.stackexchange.com/questions/18319/i-want-to-find-lim-limits-x-to-5-frac2x-25x-5-without-using-lhopital?rq=1).2012-08-18
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    This amounts to finding the second derivative of $e^x$ at $x=0$. So I guess it's important to motivate why you want to restrict methods of proof. Certainly an approach along the lines that being its own derivative characterizes $e^x$ seems viable.2012-08-18
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    What is your definition of $e^x$?2012-08-18
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    I suppose that any solution that begins by observing that $e^x= 1+x+\frac12x^2+O(x^3)$ will fail to satisfy you, even though no infinite series is being invoked?2012-08-18
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    Graph it using wolfram alpha2012-08-18
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    @ChrisEagle: I could take $e^x$ as the $\sup$ of $A=\{e^q|q\in\mathbb{Q},q\leq x\}$.2012-08-18
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    @enzotib So the next question is: what is the first property of the exponential that you do not want to use?2012-08-18
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    @enzotib: You *could* do any number of things. *Are* you doing that? If yes, what is your definition of $e$?2012-08-18
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    @JoelCohen But that would encourage people not to use l'Hopital's, which is a foolish handicap to impose on oneself.2012-08-18
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    That rule is just a make up for derivatives.2012-08-18
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    @ChrisEagle: the classical definition through the known sequence limit is ok?2012-08-18
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    I added a (weak, I know) motivation to the question.2012-08-18
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    I am not sure that your definition of $e^x$ works without a definition of $e$ - from your definition so far it could be anything.2012-08-18
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    @JoelCohen: I think the tag `alternate-proof` ought to work just fine.2012-08-18

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