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In high school calculus, the focus is on computations and not proofs. Proofs are introduced (if at all) for the sake of understanding rather than rigour.

As this answer mentions, proofs at this level make certain assumptions. For example, this answer on proving the limit of $e$ assumes $$\frac{d}{dx} \text{ln}x=\frac{1}{x}$$

And yet a search for proofs of the natural logarithm leads to those which assume knowledge of the limit.

The formulas I have seen in this context are:

  • Derivatives and Integrals of $e^x$ and $\text{ln } x$
  • Maclaurin/Taylor Series for $e^x$ and $\text{ln } x$
  • Limits involving $e$. (Expressions of the form $(1+\frac{1}{x})^x$ whose limit is e as $x\rightarrow\infty$ and their extensions/variations

What is the correct order of proof of these different results that avoids circular reasoning?

Starlight
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    There are many ways of developing the subject. It all starts with precise definitions for thing, and here the starting point varies person to person. See Spivak or Rudin for different (but logical) approaches to these matters. – peek-a-boo Nov 29 '24 at 06:56
  • I believe Spivak/Rudin are among the more difficult RA books. Would any of the easier ones also have the same results/topics? – Starlight Nov 29 '24 at 07:03
  • No, I don’t know simpler books, maybe someone else has more references; but at some stage you have to get into the real analysis technicalities because the issue here isn’t really the ‘properties’, but rather having a precise definition (which can be hard to even state without the necessary analysis background). But as an overview of different approaches, see this answer of mine (and also the accepted answer) and various sublinks (where I discuss the develoment of trigonometry starting from the ODE point of view). – peek-a-boo Nov 29 '24 at 11:31
  • Got it. Will work my way to these books. – Starlight Nov 29 '24 at 13:48
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    Velleman, Calculus: A Rigorous First Course is rigorous but easier than Spivak and Rudin. – Dan Velleman Jan 02 '25 at 22:30

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