Limits and continuity

Limits are the starting point of calculus. A limit tells us what value a function approaches as $x$ approaches a number.

In this chapter, you will learn what a limit means and how to find limits from tables, graphs, and algebra. You will also learn how limits describe continuity: whether the graph of a function has any breaks.

Limits are used in the rest of calculus. The derivative and the integral are both defined with limits.

This chapter covers the following topics:

Introduction to limits

What a limit is, limit notation, estimating limits from tables and graphs, why the value at the point does not matter, when a limit does not exist, and the precise definition of a limit (optional).

One-sided limits

Left-hand and right-hand limits, how they decide whether a limit exists, and one-sided limits of piecewise functions, absolute value, and square roots.

Computing limits

The limit laws, direct substitution, the form 0/0, and four ways to find these limits: factoring, multiplying by the conjugate, combining fractions, and expanding.

Infinite limits and limits at infinity

Limits that are infinite, vertical asymptotes, limits as x increases or decreases without bound, horizontal asymptotes, and limits at infinity of rational functions.

The Squeeze Theorem

Finding a limit by trapping a function between two others, a proof that sin x / x approaches 1, and other limits with sine and cosine.

Continuity

Continuity at a point and on an interval, removable, jump, and infinite discontinuities, which functions are continuous, and the Intermediate Value Theorem.

Before you start

This chapter uses a lot of algebra. If you need a review, these algebra pages may help:

You will also need the sine, cosine, and tangent functions, with angles measured in radians.