Lesson 7: Limits Involving Infinity

Preview

Here's the on-ramp for this lesson.


What Will We Learn?

  • In this lesson we'll finish our study of limits. Having seen earlier that sometimes the limit doesn't exist because the values of the function race off to infinity (e.g., as one approaches a vertical asymptote), in this lesson we'll delve deeper into limits involving infinity, considering also what a limit "as x approaches infinity" could mean and what information it might yield about the function.


Why Do We Need to Learn This?

  • Later in the course we'll see that certain important irrational numbers (like e, from the exponential function e^x) are defined in terms of infinite limits. Thus, infinite limits will come to form the foundation for the calculus of certain transcendental (i.e., not algebraic) functions. Furthermore, if you continue on to Calculus 2 after this course you'll see infinite limits show up in the study of sequences and series. Finally, in the sciences infinite limits show up in theoretical attempts to understand the behavior of a phenomenon after a very long time (i.e., "in the long run").

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 7.pdf

Video 1 (Examples 2.33-2.34)

Video 2 (Example 2.37)

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 7 PP.pdf