Finding Limits Via Laws and Algebra

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Anything You Want to Talk About?

When will I explain the methods and ideas you need for the problem set? When you ask me to or otherwise show that I need to through an online discussion or in-person class. But I won’t just do it because it’s the day’s topic. I’m teaching this course as what is sometimes called a “flipped class,” i.e., one in which you get the basic facts before classes through readings, videos, etc., and then we use the class time to work on example problems or otherwise address your questions — often questions come out through the examples, but you can always tell me about particular questions you want answered.

All grading happens in the individual meetings, although you don’t have to have something to grade in every such meeting. Meetings when you don’t have something ready to grade can be very useful for answering questions about a problem set, letting me know how the course is or isn’t working for you, etc. In all cases, sign up for the meetings via Google calendar; there’s a video tutorial on using it in the Sept. 2 class notes.

How do you know about problem sets and other things you’re supposed to do? I will always put things you need to do in a Canvas announcement. So watch those. If you know there’s something you want to find but aren’t sure exactly where, the “Modules” page in Canvas can help: it has almost everything produced for this course, in chronological order. Names can help you tell what kind of thing something in “Modules” is: daily class notes always start with a date, discussions always have the word “discussion,” etc.

Using Limit Laws

Key point: Limit laws let you plug in the value you’re approaching for the variable in a limit expression, as long as that doesn’t involve any undefined results e.g., division by 0.

Use limit laws to find the following limit:

Generally, for every arithmetic operation there is a limit law that lets you break up a limit based on that operation. So identify the operations, in order from last-to-do to first-to-do, and apply the related laws to break the limit down into simpler and simpler limits, until the “limit of a constant” and “limit of x” laws give you actual numbers:

Normally you wouldn’t write out a result using the limit laws in this much detail. Instead, you’d use the “key point” above to plug the value being approached in for the variable in the limit and evaluate everything at once. The main value of the limit laws is that they give you precise reasons to justify doing this. And that’s an important point about math in general: everything it does has a reason (unless it’s a definition or so-called “axiom”).

Using Algebra

In many cases, you can’t actually use the limit laws immediately, because doing so would involve an illegal division, often 0/0.

For example, what happens if you try to use only the limit laws to find lim_{t → 0} (t² + 2t) / t? You quickly get to having to evaluate 0 / 0. (And even sooner to a place where the limit law for quotients doesn’t apply, because the denominator is 0.)

But if you simplify (t² + 2t) / t first, you can get around this problem:

Next

More examples of using algebra to simplify and find limits.

Review the reading on “Additional Limit Evaluation Techniques” in section 2.3 of the book.

Keep following and contributing to the online discussion of algebraic methods of finding limits.

Friday’s class is a Cohort B class.

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