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According to this source (and many others):

If $r$ is a positive rational number and $c$ is any real number, then

$$\lim_{x\to \infty} \frac{c}{x^r} = 0$$

Consider some basic limit problem, for example:

$$\lim_{x\to \infty} 5 + \frac{1}{x}$$

If I get this correctly, it solves like this: $$5 + 0 = 5$$

And now coming back to my main question:

$$\lim_{x \to \infty} (1 + \frac{1}{x})^x$$

Since $\lim_{x\to \infty} \frac{1}{x} = 0$, then

$$\lim_{x \to \infty} (1 + \frac{1}{x})^x = (1+0)^{\infty}$$

Now, although I'm not sure what $(1)^{\infty}$ means, but it definitely doesn't look like anything close to 2.71828

So the question is, why $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$?

I understand that you can just plug large numbers into $(1 + \frac{1}{x})^x$ and see that result approaches ~2.71828, but I'm more interested in mathematical derivation.

cmk
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Stokolos Ilya
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  • why are so many people saying "since ..., then ..."?? i don't know where that started, but it's so wrong and sounds so weird – mathworker21 Aug 10 '19 at 14:39
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    $\ln((1+\frac{1}{x})^x) = x\ln(1+\frac{1}{x}) \sim 1$ – mathworker21 Aug 10 '19 at 14:40
  • @mathworker 21 https://english.stackexchange.com/questions/506491/what-is-the-difference-between-these-two-sentences-if-a-is-true-then-b-is-tru – Stokolos Ilya Aug 10 '19 at 14:42
  • @mathworker21 According to JohnLawler, although it is uncommon, but not grammatically incorrect. – Stokolos Ilya Aug 10 '19 at 14:43
  • @Nelver I don't see where JohnLawler says it is grammatically correct – mathworker21 Aug 10 '19 at 14:48
  • @Nelver https://english.stackexchange.com/questions/266305/the-since-then-construction – mathworker21 Aug 10 '19 at 14:49
  • Quoting first sentence from his post: "To answer the second question first, then is optional after if; also after since, but not as common.". "Not as common" means (implicitly) that, although it is rarely used, but it is still correct. Otherwise, if it is wrong and is not used anywhere, why wouldn't he explicitly say so? – Stokolos Ilya Aug 10 '19 at 14:52
  • "Then is optional after since". Meaning, again, that it can be used. – Stokolos Ilya Aug 10 '19 at 14:53
  • Please take comments related to grammar to the relevant posts, not here. – Simply Beautiful Art Aug 10 '19 at 14:58
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    Think of it as a race. On the one hand, you have (1 + N), where N is getting smaller and smaller and smaller. But, you are exponentiating it higher and higher. While $1^\infty$ is in fact $1$, that is not the same thing that is happening here. It is 1 plus a super-duper-teeny-tiny amount. That amount would be negligible if it were raised to a finite power. However, here we are raising that teeny-tiny amount to an INFINITE power. Therefore, even though it is infinitely small, the infinite power magnifies its affect infinitely. $e$ is just where it settles in the race between $\infty$ and 0. – johnnyb Aug 10 '19 at 19:41

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