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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 Lee Yiyuan, Brian Fitzpatrick, Claude Leibovici, Magdiragdag, Sujaan Kunalan Mar 30 at 7:15

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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What would you call fascinating? Limits aren't very fascinating... –  Shahar Mar 29 at 21:08
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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
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Let $\pi(x)$ be the number of primes $\le x$. Then $\lim_{x\to\infty} \frac{\pi(x)}{x/\ln x}=1$. –  André Nicolas Mar 29 at 21:13
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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
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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

10 Answers 10

up vote 3 down vote accepted

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

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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
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@Shahar Not in the convential definition of convergence though. –  Christoph Mar 29 at 21:42

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

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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

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

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

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

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

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

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

enter image description here

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

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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

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