# Limits

Contents

Limit introduction, squeeze theorem, and epsilon-delta definition of limits

Much of calculus deals with the idea of infinity. You will commonly hear people speaking about something "infinitely small", for example. Limits are the tool underlying all calculus which let us talk sensibly about infinity. In particular, they give us the language to talk about being "infinitely close" to some number by considering what happens as you approach that number.

In this tutorial, we will build our ability to visualize limits by estimating them based on graphs of functions. We will look at both one-sided and two-sided limits.

We often attempt to find the limit at a point where the function itself is not defined. In this tutorial, we will use algebra to "simplify" functions into ones where it is defined. Given that the original function and the simplified one may be identical except for the limit point in question, this is a useful way of finding limits.

A function isn't continuous when there is a "break" in its graph. This tutorial uses limits to define this idea more formally and gives practice thinking about continuity (and discontinuity) in terms of limits.

You have a basic understanding of what a limit is. Now, in this tutorial, we can explore situation where we take the limit as x approaches negative or positive infinity (and situations where the limit itself could be unbounded).

If a function is always smaller than one function and always greater than another (i.e. it is always between them), then if the upper and lower function converge to a limit at a point, then so does the one in between. Not only is this useful for proving certain tricky limits (we use it to prove lim (x → 0) of (sin x)/x, but it is a useful metaphor to use in life (seriously). :)
This tutorial is useful but optional. It is covered in most calculus courses, but it is not necessary to progress on to the "Introduction to derivatives" tutorial.

This tutorial introduces a "formal" definition of limits. So put on your ball gown and/or tuxedo to party with Mr. Epsilon Delta (no, this is not referring to a fraternity).
This tends to be covered early in a traditional calculus class (right after basic limits), but we have mixed feelings about that. It is cool and rigorous, but also very "mathy" (as most rigorous things are). Don't fret if you have trouble with it the first time. If you have a basic conceptual understanding of what limits are (from the "Limits" tutorial), you're ready to start thinking about taking derivatives.