Lesson 5: Two-Sided Limits; Continuity

Preview

Here's the on-ramp for this lesson.


What Will We Learn?

  • In this lesson we'll continue our limits adventure and learn about two-sided limits and continuity. Unlike one-sided limits, two-sided limits investigate what happens to a function's y-values as we approach x=c from both the left and the right. When each of those one-sided limits exists, their values are the same, and this value is equal to f(c), then we call the function continuous at x=c. As we'll learn, the graphs of continuous functions have no breaks or holes (example: any polynomial).


Why Do We Need to Learn This?

  • Continuous functions model many of the real-world phenomena we care to model. Likewise, continuous functions are the most common types of functions we care to differentiate and integrate (operations we'll study later in the course). And because continuity is defined in terms of one- and two-sided limits, we'll start this lesson by first learning about two-sided limits, which describe the behavior of a function's values as the input approaches a specified number from both directions on the number line (left and right).

Review

Learn

The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:

  • I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.

Lesson Notes

Lesson 5.pdf

Video 1 (Example 2.5)

Video 2 (Example 2.6)

Video 3 (Example 2.7)

Video 4 (Examples 2.9-2.10)

Video 5 (Example 2.11)

Reflect

If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:

  • Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.

  • Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.

If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:

  • Please summarize the main mathematical takeaways from the lesson notes.

  • What was the most interesting part of what you learned, and why?

  • What, if anything, do you still find confusing?

Practice

Work through the practice problems suggested below to see how much of this lesson you've understood.

Lesson 5 PP.pdf