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Might be a silly question, or a complete lack of intuition from me, but I cannot figure out how to properly solve limits using some trivial variable change.

The example that triggered me is the limit you obtain while differentiating $e^x$:

$$\lim_{h \to 0} \frac{e^h - 1}{h} = 1$$

From here, one could introduce the variable change $u = e^h - 1 \implies h = \ln(u + 1)$ and then obtain $\lim_{u \to 0} \frac{u}{\ln(u + 1)} = 1$. Now, one could again multiply both the numerator and denominator by $\frac{1}{u}$:

$\lim_{u \to 0} \frac{u}{\ln(u + 1)} \cdot \frac{\frac{1}{u}}{\frac{1}{u}} = \lim_{u \to 0} \frac{1}{\ln(u + 1) \cdot \frac{1}{u}} = 1$

I am just trying to understand if there is an underlying formal framework behind those substitutions or it only involves intuitive pattern recognition.

emandret
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1 Answers1

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It is mainly a matter of tricky pattern recognition.

Usually the foundamental result proved as first step is that as $x \to \infty \implies \left(1+\frac1x\right)^x\to e$ and from here we can prove that both

$$\lim_{h \to 0} \frac{e^h - 1}{h} = 1 \quad \lim_{h \to 0} \frac{\log (1+h)}{h} = 1$$

which are indeed equivalent.

user
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  • Is it possible to find the first formula without involving it in the limit calculation? From a formal pov – emandret Oct 31 '19 at 09:45
  • @gatosec Yes I think it is possible also prove indipendently that $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$. Maybe there is something on MSE on that. – user Oct 31 '19 at 09:51
  • For example here or also here. – user Oct 31 '19 at 09:52
  • but the power series is basically the binomial expansion of the Bernoulli formula, this is circular in a sense. – emandret Oct 31 '19 at 09:55
  • @gatosec The second link contains an argument completely independent. – user Oct 31 '19 at 09:56
  • Oh sorry. Was not updated here on my browser. – emandret Oct 31 '19 at 09:56
  • what’s interesting is that the Bernoulli formula is not directly related to the limit in my post, we usually say it’s e because we know it’s e in the first place, but how can you connect this limit definition to the number e without involving its very own definition? The posts you linked don’t really addressed this issue. – emandret Oct 31 '19 at 10:08
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    @gatosec : there are multiple approaches (and none of them being circular) to the theory of $e, \exp, \log$ and each approach has to start from somewhere. You should explicitly mention your starting point. – Paramanand Singh Oct 31 '19 at 17:58
  • @ParamanandSingh, after reading your blog, I had the answer to most of my questions. At least, regarding the theory behind the natural logarithm and direct consequences about differentiating exponentiation. However, I still do not understand how to handle limits properly, I lack fundamentals skills and I am prone to do invalid algebraic manipulations, especially when rewriting them. – emandret Oct 31 '19 at 18:44
  • In my OP, I mentioned the variable change $u = e^h - 1$, what is the formal basis behind this? Is there any rule of what variable change one should consider when encountering such limits? Or is it just invented out of thin air? – emandret Oct 31 '19 at 18:46
  • @gatosec: this is called rule of substitution for limits. You can see https://math.stackexchange.com/a/1073047/72031 – Paramanand Singh Nov 01 '19 at 01:46
  • @gatosec: I have also written about limits in my blog. You may have a look at those posts starting from this one. – Paramanand Singh Nov 01 '19 at 01:48