Chapter 1 · Introduction

Chapter 1 — Limits and Continuity

Approaching without arriving — the single idea that the rest of calculus is built on.

The idea of a limit is central to all of calculus. In this chapter we describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell whether a function does or does not have a limit at a particular value. The chapter is informal and intuitive — the aim is to build a picture you can trust.

Why start here? A derivative is a limit of slopes; an integral is a limit of sums; continuity is defined by a limit. Master limits and the rest of the course is built on solid ground.

Tables & graphs→Limit laws & algebra→Continuity→Behaviour at $\infty$
🎯 By the end of this chapter you can…
  • estimate limits from tables and graphs, including one-sided limits;
  • evaluate limits with the limit laws, factoring, conjugates and simplification;
  • decide whether a function is continuous at a point or on an interval;
  • find limits at infinity and the horizontal and vertical asymptotes of a graph.
Teaser — Zooming in on a Hole

$f(x)=\dfrac{x^2-4}{x-2}$ is undefined at $x=2$. Slide $x$ toward $2$ from either side and watch $f(x)$.

$x$—
$f(x)$—
distance from 4—
The values close in on $4$, even though $f(2)$ does not exist: $\displaystyle\lim_{x\to2}f(x)=4$. That is the whole idea of a limit — it is about approaching, not arriving. Start 1.1 →

Sections in this chapter

1.1

The Limit of a Function

Estimate limits with tables and graphs; one-sided limits and when a limit fails to exist.

$\displaystyle\lim_{x\to a}f(x)=L$
Open section →1.2

The Limit Laws

Compute limits exactly with the limit laws, factoring, conjugates and complex fractions.

$\tfrac00\Rightarrow$ factor, cancel, substitute
Open section →1.3

Continuity

The three-condition test for continuity at a point and continuity on intervals.

$\displaystyle\lim_{x\to a}f(x)=f(a)$
Open section →1.4

Limits Involving Infinity

Limits at $\pm\infty$, infinite limits, horizontal and vertical asymptotes.

$\displaystyle\lim_{x\to\infty}\tfrac1x=0$
Open section →
✅ Before you start
Comfortable with factoring, conjugates and domains? If not, spend ten minutes in 0.1 §6 Factoring and fractions and §2 Functions.
← 0.1 Pre-Calculus Review1.1 The Limit of a Function →