Fascinating limits? (highschool)

Question

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.

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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....

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

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

Answer 10

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