Here's the on-ramp for this lesson.
What Will We Learn?
In this lesson we'll revisit limits in the context of transcendental functions. For these functions, we'll learn about one- and two-sided limits, calculating limits algebraically, and working with limits involving infinity.
Why Do We Need to Learn This?
The base of natural exponential function e^x, the irrational number $e$ (Euler's number), is defined in terms of an infinite limit (see Calculus Simplified, pg. 32). Thus, most exponential and logarithmic functions -- and their real-world applications -- have infinite limits built in to their very definition. In terms of trigonometric functions, the particular limits we'll learn in this lesson will form the foundation for differentiating trigonometric functions in the next lesson.
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.
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?
Work through the practice problems suggested below to see how much of this lesson you've understood.