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diff --git a/mindmap/limit.org b/mindmap/limit.org new file mode 100644 index 0000000..946e4d2 --- /dev/null +++ b/mindmap/limit.org @@ -0,0 +1,48 @@ +:PROPERTIES: +:ID: 122fd244-ffeb-47d0-89ce-bf9bc6f01b70 +:END: +#+title: limit +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +A limit in mathematics is a tool used to describe the intuitive process +of a value or a set of values tending towards another. First, we will define +limits as they pertain to sequences, and then we will define them on [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][functions]]. +For a sequence $\{s_{n}\}$: + +\begin{align*} +\lim s_{n} = s \iff \forall \epsilon > 0, \exists N , n > N \implies | s_{n} - s | < \epsilon +\end{align*} + +What this means is that at some point in the sequence, for some choice of epsilon, no matter how small +it is, there has to be an index where every term after that index is closer to $s$ than epsilon. If +some single number $s$ and sequence $\{s_{n}\}$ fulfills this criteria, then it is said that the limit +of the sequence is $s$. Generally speaking, we use the set $\mathbb{R} \cup \{ -\infty, +\infty \}$, where there is a natural +ordering: + +\begin{align*} +\forall a \in \mathbb{R}, - \infty < a < +\infty +\end{align*} +defined. Note that we can define equivalence relations on these symbols, but algebra reamins undefined. +** Limits on Monotone Sequences +An increasing sequence is a sequence $\{s_{n}\}$ defined such that: +\begin{align*} +\forall n \in \mathbb{N}, \forall m \in \mathbb{N}, n \ge m \implies s_{n} \ge s_{m}. +\end{align*} +and now we wish to prove that the limit of monotone sequences always exist. + +\begin{align*} +\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ +\end{align*} + +#+begin_theorem +If I am bad, then you are too. +#+end_theorem + +#+begin_proof + +#+end_proof diff --git a/mindmap/ltximg/org-ltximg_02aa3ea174cc91456ea23713ffe95af053dcb277.png b/mindmap/ltximg/org-ltximg_02aa3ea174cc91456ea23713ffe95af053dcb277.png Binary files differnew file mode 100644 index 0000000..78e0534 --- /dev/null +++ b/mindmap/ltximg/org-ltximg_02aa3ea174cc91456ea23713ffe95af053dcb277.png diff --git 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index ac1c5d2..54ab922 100644 --- a/mindmap/partial derivative.org +++ b/mindmap/partial derivative.org @@ -11,4 +11,4 @@ * Definition The operator \(\frac{\partial}{\partial x_{n}}\) takes the [[id:31d3944a-cddc-496c-89a3-67a56e821de3][derivative]] of a function \(f(\vec{x})\) with respect to \(x_{n}\) in the direction of the \(x_{n}\) axis. In other words, we take the derivative \(f(\vec{x})\) as if all variables -except \(x_{n}\) is constant. +except \(x_{n}\) is constant; this is useful in cases where you only care about one variable. |
