This guide covers discontinuities in AP Calculus AB/BC, including removable (holes), jump, and asymptote discontinuities. It explains how to identify each type, focusing on limits and piecewise functions. Examples, practice questions, and exam tips are provided to help prepare for the AP exam.

#AP Calculus AB/BC: Types of Discontinuities - Your Ultimate Guide 🚀

Hey there, future AP Calculus master! Let's dive into the world of discontinuities. This guide is designed to be your go-to resource, especially the night before the exam. We'll make sure everything clicks and you feel confident. Let's get started!

This topic is crucial for understanding function behavior and is frequently tested on both multiple-choice and free-response questions. Understanding discontinuities is key to mastering limits and continuity.

#Introduction to Discontinuities

First, let's nail down what a discontinuity actually is. Remember, for a function to be continuous, three things must be true:

It must be defined at all x values in its domain.
The limit must exist at all points in its domain.
The limit from the left must equal the limit from the right at all points in its domain.

If any of these rules are broken, you've got a discontinuity! Think of it like this: if you have to lift your pencil while drawing a graph, you've hit a discontinuity. ✏️

#Removable Discontinuities (aka "Holes")

These are the simplest type of discontinuity, often called "holes" in the graph. They happen when a single point is out of place. 🕳️

Key Concept

Removable discontinuities are crucial because they often involve algebraic manipulation (factoring) to identify and 'remove' them. This skill is often tested.

There are two main reasons for removable discontinuities:

"Blips" in the function, often caused by piecewise definitions.
Common factors in rational functions (fractions).

#Example 1: Piecewise "Blip"

Consider this piecewise function:

$y = \begin{cases} x^2 & \text{if } x < 1 \\ x - 1 & \text{if } x = 1 \\ -x + 2 & \text{if } x > 1 \end{cases}$

Here's what it looks like:

Removable Discontinuity Example 1

Caption: Notice the "hole" at x=1. Th...

#AP Calculus AB/BC: Types of Discontinuities - Your Ultimate Guide 🚀

#Introduction to Discontinuities

First, let's nail down what a discontinuity actually is. Remember, for a function to be continuous, three things must be true:

It must be defined at all x values in its domain.
The limit must exist at all points in its domain.
The limit from the left must equal the limit from the right at all points in its domain.

If any of these rules are broken, you've got a discontinuity! Think of it like this: if you have to lift your pencil while drawing a graph, you've hit a discontinuity. ✏️

#Removable Discontinuities (aka "Holes")

These are the simplest type of discontinuity, often called "holes" in the graph. They happen when a single point is out of place. 🕳️

Key Concept

Removable discontinuities are crucial because they often involve algebraic manipulation (factoring) to identify and 'remove' them. This skill is often tested.

There are two main reasons for removable discontinuities:

"Blips" in the function, often caused by piecewise definitions.
Common factors in rational functions (fractions).

#Example 1: Piecewise "Blip"

Consider this piecewise function:

$y = \begin{cases} x^2 & \text{if } x < 1 \\ x - 1 & \text{if } x = 1 \\ -x + 2 & \text{if } x > 1 \end{cases}$

Here's what it looks like:

Removable Discontinuity Example 1

Caption: Notice the "hole" at x=1. Th...

For a function to be continuous at a point, which of the following conditions MUST be true? 🤔

The function must be defined at that point, and the limit must exist

The function must be defined at that point, and the left-hand limit must equal the right-hand limit

The function must be defined at the point, the limit must exist at the point, and the limit must equal the function's value at that point

The function must be differentiable at that point

For a function to be continuous at a point, which of the following conditions MUST be true? 🤔

The function must be defined at that point, and the limit must exist

The function must be defined at that point, and the left-hand limit must equal the right-hand limit

The function must be defined at the point, the limit must exist at the point, and the limit must equal the function's value at that point

The function must be differentiable at that point

Exploring Types of Discontinuities

Hannah Hill

#AP Calculus AB/BC: Types of Discontinuities - Your Ultimate Guide 🚀

#Introduction to Discontinuities

#Removable Discontinuities (aka "Holes")

#Example 1: Piecewise "Blip"

How are we doing?

Exploring Types of Discontinuities

Hannah Hill

#AP Calculus AB/BC: Types of Discontinuities - Your Ultimate Guide 🚀

#Introduction to Discontinuities

#Removable Discontinuities (aka "Holes")

#Example 1: Piecewise "Blip"

How are we doing?