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Prove the following limit relations: $$\lim_{x\to0} (1+x)^{1/x} = e$$ $$\lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^n = e^x$$

I'm not sure how to prove this as I'm not really sure what tools I have to prove it. I know by definition that the two limit relations are true, but any advice as to how to solve this specific problem/similar problems would be very appreciated!

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    $\begingroup$ Define $e$ and define $e^x$. These limits are often given as the definitions. $\endgroup$ – Mark Viola Mar 28 '17 at 3:12
  • $\begingroup$ @Dr.MV I don't think my professor wants us to just give the definitions, though - he wants a technical proof of some sort. $\endgroup$ – mizichael Mar 28 '17 at 3:14
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    $\begingroup$ The question is "How does one define $e$?" Without a starting definition, one cannot proceed. $\endgroup$ – Mark Viola Mar 28 '17 at 3:15
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    $\begingroup$ @mizichael You wrote I know by definition that the two limit relations are true but also ask how to prove this. You don't/can't prove definitions, so what's being asked here is what are the actual definitions you work with, and what remains to be proved using those definitions. $\endgroup$ – dxiv Mar 28 '17 at 3:18
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    $\begingroup$ I don't know which characterization your professor has chosen implicitly. Have you covered Taylor series? $\endgroup$ – Mark Viola Mar 28 '17 at 3:24
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If we let x=1 in the second limit is very easy to prove they are the same limit, you make a change of variable n=1/x, as x approaches 0, n tends to infinity, so you just replace in the limit and they both tend to e

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