# Limits A simple but useful example of a limit is as follows. Consider a straight line going from $0$ to $1$. We want to know the length of this line. Of course we can immediately see that it is simply $1-0=1$, but what if we wanted to calculate it using a limit? We could do so by breaking the line up into segments and then adding up the length of the individual segments. If we started with $n=2$ segments, we would have: $\frac{1}{2} + \frac{1}{2} = 1$ If we then let $n$ increase, what would happen? Well, we would have smaller and smaller segments, but we would have more of them: $ \overbrace{ \frac{1}{n} + \frac{1}{n} + \dots + \frac{1}{n} }^{n \text{ segments}} = 1$ The concept of the limit allows us to quantify what happens as we let $n$ approach $\infty$: $\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{n}= 1$ We can summarize this cleanly as follows: > If you divide the unit interval [0,1][0,1] into N equal parts where N is an infinite number, each of the intervals of the subdivision will be of infinitesimal length. Thus infinitely many subintervals can indeed add to a finite length, but those subintervals can't have appreciable length: they must be infinitesimal. The key idea to keep in mind here is that we truly do end up with segments that are infinitesimally small, but we have infinitely many of them. This is directly tied to the idea of arbitrarily large and arbitrarily small (see the idea of always being able to pick a number larger or smaller than your challenger-express in Vector Calculus, Linear Algebra, and Differential Forms). Even better, see the [**epsilon-delta** definition](https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/) of a limit. --- Date: 20210723 Links to: [High Dimensional Spaces](High%20Dimensional%20Spaces.md) (see the whiteboard photos here) Tags: References: * [What are infinitesimals?](https://math.stackexchange.com/questions/771952/what-is-the-answer-to-the-paradox-of-the-infinitesimal) * [Epsilon Delta, limit definition](https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/)