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.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.

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.