AP Calculus AB Unit 1: Limits and Continuity

The Big Picture

Welcome to the starting line of calculus! This unit introduces the single most important idea in the entire course: the limit. Think of it this way: Algebra is great at finding values at specific points, like the speed of a car averaged over an hour. But what about the car's exact speed at the very instant you look at the speedometer? That's a question of instantaneous change, and algebra alone can't answer it.

Limits are the mathematical tool that bridges this gap. A limit describes the value a function approaches as you get infinitely close to a certain input, without necessarily caring about what happens at that exact input. This powerful idea of "approaching" is the engine that drives everything else in calculus, from finding the slope of a curvy line to calculating the area of a complex shape. Mastering limits now will build a rock-solid foundation for every unit to come.

Key Questions

As you work through this unit, keep these fundamental questions in mind. If you can answer them by the end, you're in great shape.

• How can we describe what's happening to a function's output value as its input gets closer and closer to a specific number?

• What does it mean, mathematically, for a function's graph to be an "unbroken" curve, and how can we test for and classify any breaks?

• How do limits help us understand a function's long-term behavior and identify key graphical features like asymptotes?

• If a function is continuous (unbroken), what can we guarantee about the values it must take on between any two points?

Your Learning Path

This unit is structured to build your understanding from the ground up, moving from the concept of a limit to its calculation and finally to its major application: continuity.

1. The Concept of the Limit

Topic 1.1 - 1.4: Introducing, Defining, and Estimating Limits

You'll start by building an intuitive understanding of what a limit is. You will learn the formal notation for writing limits and practice estimating a limit's value by analyzing the behavior of a function on a graph or in a table of values. This section is all about what a limit is before we get into how to calculate it.

2. The Calculation of Limits

Topic 1.5 - 1.7: Determining Limits with Properties and Manipulation

Here, you'll move from estimating to calculating exact values for limits. You will learn the fundamental properties of limits and master the essential algebraic techniques—like factoring, rationalizing, and simplifying complex fractions—that are required when direct substitution doesn't work.

Topic 1.8: Determining Limits Using the Squeeze Theorem

You will learn a special and powerful theorem for finding the limit of a function that is "squeezed" between two other functions whose limits are known and equal.

Topic 1.9: Connecting Multiple Representations of Limits

This is a capstone topic for the first half of the unit. You'll practice synthesizing your knowledge by moving fluidly between graphical, numerical, and analytical representations of limits.

3. Continuity and Its Consequences

Topic 1.10 - 1.13: Defining, Analyzing, and Fixing Discontinuities

This section introduces continuity, the first major application of limits. You will learn the three-part formal definition of continuity at a point, use it to identify and classify different types of "breaks" (discontinuities) in a graph, and even learn how to "fix" certain types of discontinuities.

Topic 1.14 - 1.15: Connecting Limits to Asymptotes

You will explore the connection between limits and the graphical behavior of functions. You'll see how infinite limits (where the function's output flies up to infinity or down to negative infinity) correspond to vertical asymptotes, and how limits at infinity (where the input gets infinitely large) correspond to horizontal asymptotes.

Topic 1.16: Working with the Intermediate Value Theorem (IVT)

You'll learn your first major theorem of calculus. The IVT is a powerful consequence of continuity that allows you to guarantee the existence of a particular function value on an interval, even without knowing exactly where it is.

How to Succeed in This Unit

• Master Limit Notation. On the AP Exam, notation is not optional—it's part of the answer. Always write the full lim expression, including the x → c part, in every step of your work. Do not drop the limit notation until you have actually substituted the value.

• Show Your Algebraic Work. When a limit cannot be found by direct substitution, you must show the algebraic manipulation that leads to your answer. Simply writing down the problem and then the final answer will not earn full credit. Show the factoring, the cancellation, or the rationalization.

• Memorize the 3-Part Definition of Continuity. Many exam questions ask you to justify why a function is or is not continuous at a point. To earn full credit, you must explicitly check all three conditions: the point exists, the limit exists, and they are equal.

• Be Specific About "DNE". While a limit that approaches different values from the left and right "Does Not Exist" (DNE), a limit that approaches infinity is also technically DNE. Be more descriptive. If a limit is infinite, write ∞ or -∞ as the answer. This demonstrates a deeper understanding and is often required for full credit.