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Welcome to Unit 2 Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together. A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct. You can start by reviewing the unit learning outcomes and then reviewing the unit resources.
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Source and License: This work is licensed by Saylor Academy under a Creative Commons Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA 4.0). This content was created using Genially and Synthesia. AI-generated avatars and voices in this video were created using Synthesia and remain subject to Synthesia’s Terms of Service; these elements are not covered by the Creative Commons license. Synthesia trademarks and services remain the property of Synthesia. All Genially proprietary elements such as templates, themes, built-in assets, stock media, and other “Genially Content” remain subject to Genially’s Terms of Service and are not covered by this Creative Commons license. These elements must remain embedded in the course and cannot be reused or redistributed independently.
Source and License: This work is licensed by Saylor Academy under a Creative Commons Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA 4.0). This content was created using Genially and Synthesia. AI-generated avatars and voices in this video were created using Synthesia and remain subject to Synthesia’s Terms of Service; these elements are not covered by the Creative Commons license. Synthesia trademarks and services remain the property of Synthesia. All Genially proprietary elements such as templates, themes, built-in assets, stock media, and other “Genially Content” remain subject to Genially’s Terms of Service and are not covered by this Creative Commons license. These elements must remain embedded in the course and cannot be reused or redistributed independently.
AI Summary
Unit 2 introduces functions, limits, and continuity, which form the foundation of calculus. Here are some key takeaways:
- Understand the concept of a limit and why it is central to calculus.
- Explore continuity and how it affects mathematical outcomes.
- Recognize how limits and continuity work together in solving calculus problems.
You can start by reviewing the unit learning outcomes and the unit resources.
Unit 2 Introduction Video
Saylor Academy
Created on February 11, 2026
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Transcript
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Experiencing playback issues or need translation options?
Welcome to Unit 2 Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together. A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct. You can start by reviewing the unit learning outcomes and then reviewing the unit resources.
AI Summary
Click on the icons to access a summary of this page or the video transcript
Video Transcript
Source and License: This work is licensed by Saylor Academy under a Creative Commons Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA 4.0). This content was created using Genially and Synthesia. AI-generated avatars and voices in this video were created using Synthesia and remain subject to Synthesia’s Terms of Service; these elements are not covered by the Creative Commons license. Synthesia trademarks and services remain the property of Synthesia. All Genially proprietary elements such as templates, themes, built-in assets, stock media, and other “Genially Content” remain subject to Genially’s Terms of Service and are not covered by this Creative Commons license. These elements must remain embedded in the course and cannot be reused or redistributed independently.
Source and License: This work is licensed by Saylor Academy under a Creative Commons Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA 4.0). This content was created using Genially and Synthesia. AI-generated avatars and voices in this video were created using Synthesia and remain subject to Synthesia’s Terms of Service; these elements are not covered by the Creative Commons license. Synthesia trademarks and services remain the property of Synthesia. All Genially proprietary elements such as templates, themes, built-in assets, stock media, and other “Genially Content” remain subject to Genially’s Terms of Service and are not covered by this Creative Commons license. These elements must remain embedded in the course and cannot be reused or redistributed independently.
AI Summary
Unit 2 introduces functions, limits, and continuity, which form the foundation of calculus. Here are some key takeaways:
- Understand the concept of a limit and why it is central to calculus.
- Explore continuity and how it affects mathematical outcomes.
- Recognize how limits and continuity work together in solving calculus problems.
You can start by reviewing the unit learning outcomes and the unit resources.