# Fascinating limits? (highschool) [closed]

I wonder if there is someone who knows any cool limit who they're are willing to share. I have just started using them in highschool and is interested in learning more.

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## closed as primarily opinion-based by Yiyuan Lee, Brian Fitzpatrick, Claude Leibovici, Magdiragdag, Sujaan KunalanMar 30 at 7:15

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What would you call fascinating? Limits aren't very fascinating... –  Shahar Mar 29 at 21:08
Depending on your age (I'd say at least 17 for most bright mathematicians) GH Hardy's "Pure Mathematics" is a decent source of limits and their possibilities. –  Mark Bennet Mar 29 at 21:10
Let $\pi(x)$ be the number of primes $\le x$. Then $\lim_{x\to\infty} \frac{\pi(x)}{x/\ln x}=1$. –  André Nicolas Mar 29 at 21:13
I would strongly disagree with the statement that limits aren't fascinating, as all of mathematical analysis has been called the "art of taking limits." Though not as interesting as Andres example, $\lim_{n\rightarrow\infty}\frac{(1+1/n)^{n^2}}{e^n}=1/\sqrt{e}$. –  user88849 Mar 29 at 21:14
FYI, G. H. Hardy's A Course of Pure Mathematics is available gratis (typeset in LaTeX, with modernized notation and re-created diagrams) from Project Gutenberg, or in HTML from the Sayahna Foundation. –  user86418 Mar 29 at 22:21

$$\lim_{n\to\infty}\sum_{k=1}^n\frac1{k^2}=\frac{\pi^2}{6}$$

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Oh, if series count, then: $$\lim_{n\to\infty}\sum_{k=1}^n k =-\frac{1}{12}$$ (according to my string theory book) –  Shahar Mar 29 at 21:36
this one is really fascinating :O –  user3346607 Mar 29 at 21:41
@Shahar Not in the convential definition of convergence though. –  Christoph Mar 29 at 21:42

1) The epxonential function $f(x)=e^{-x}$. Here take the limit $x \to \infty$

2) The function $f(x)=\frac{1}{x}$. Here take the limit $x \to 0$ and $x \to \infty$

3) The function $f(x)=\frac{\sin x}{x}$. Here take the limit $x \to 0$

4) And (sorry to disappoint you), but things some times do not converge and they oscillate for ever....

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Let $\pi(x)$ be the number of primes $\le x$. Then $$\lim_{x\to\infty}\frac{\pi(x)}{x/\ln x}=1.$$ This is the famous Prime Number Theorem.

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$$\lim_{n\to\infty} \left(1+\frac x n\right)^n=e^x=\lim_{n\to\infty} \sum_{k=0}^n \frac{x^k}{k!}$$

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$$\lim_{x\to0}\frac{\sin x}{x}=1$$

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It might be worth mentioning here - as OP is a high-school student - that x here is expressed in radians, rather than degrees. –  Aky Mar 30 at 7:01

Falls out of Stirling's approximation, but it's still cool: $$\lim_{n \to \infty} \frac{\ln n!}{n \ln n} = 1$$

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$$\lim_{n\to\infty}\underset{\sum_{k=1}^nx_k^2\le1}{\int\ldots\int}1\ dx_1\ldots dx_n=0$$

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My personal favorite: $\lim_{n \to \infty} (1+\frac{1}{n})^n = e$

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I'm surprised nobody posted this yet: $$\lim_{n\to\infty} \left( 1+\frac{i\pi}{n} \right)^n = -1$$ Or: $$\lim_{n\to\infty} \sum_{k=0}^{n} \frac{(i\pi)^k}{k!} = -1$$

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Well, technically this is just a special case of Cristoph's answer... –  Mario Carneiro Mar 30 at 2:19

$f(x)=x$
$\lim_{n \to a}f(x)=a$.
simple. but construct compution of limit of polynomials

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