# Limits

Contents

Preview of the calculus topic of limits

## Limits basics

Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can think about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to a curve.
If you have a decent background in algebra (graphing and functions in particular), you'll hopefully enjoy this tutorial!

Intro to limits

What are limits?

Limits intuition

Introduction to the intuition behind limits

## Estimating limits from graphs

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.

One-sided limits from graphs

Here we use graphs to find the limit of a function as you approach a point from just one side.

One-sided limits from graphs

Practice finding limits by looking at graphs.

Two-sided limits from graphs

Another example of a two sided limit at a discontinuous point.

Two-sided limit at a point of discontinuity

The limit does not always exist. Here we see an example of when this happens.

Two-sided limits from graphs

Practice finding two sided limits by looking at graphs.

## Estimating limits numerically

Estimate limits numerically, for example by filling out a table of values near a specific value of x.

Estimating limit numerically

Here we walk through a specific practice problem estimating the limit of a function using numerical data.

Finding limits numerically

Find limits numerically by completing a table of values.

## Finding limits algebraically

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.

Finding limits algebraically

In this example we find the limit of a function near an undefined point by simplifying the function near that point.

Limit properties

What is the limit of the sum of two functions? What about the product?

Two-sided limits using algebra

Practice finding two-sided limits by simplifying functions algebraically.

Two-sided limits using advanced algebra

Watch Sal find a tricky limit by factoring a cubic expression.

Two-sided limits using advanced algebra

Practice finding two-sided limits using advanced algebra