Limit Laws Discussion

This is the first of several discussions we’ll use to start understanding algebraic ways of finding limits, i.e., ways of simplifying expressions until you can use standard “limit laws” to find their limits. This discussion focuses on tips for reading math and on the limit laws; future ones will work more on simplifications.

Please make one or more posts to this discussion by class time on Wednesday, September 7, responding to one or more of the following....

Reading Math

The reading on finding limits algebraically (“Evaluating Limits with the Limit Laws,” “Limits of Polynomial and Rational Functions,” and “Additional Limit Evaluation Techniques” in section 2.3) is to my mind the hardest reading I’ve asked you to do so far, and reading math isn’t easy in any case. Here are some of my own tricks for reading math, or other academic/technical text:

• Read several times. In my case, I usually read something once to get the gist of it, and then go back to specific parts for more careful reading when I need to use the ideas from those parts.
• Stop often to check your understanding. I will pause after key sentences, definitions, equations, etc. to ask myself “what does this mean,” to try to paraphrase the ideas in my own words. This may well mean pausing after nearly every sentence, or every line of an example.
• Visualize. As part of paraphrasing ideas for myself, I sometimes find it helpful to imagine a mental picture of what’s going on (or a real picture on paper if you want).
• Think of examples. Another good way to be sure you understand key ideas is to make up a simple example of your own.
• Read the examples. Even if you think you understand an idea, read through the book’s examples of it, asking yourself “why did they do this” and “how does this follow from the previous steps” at every step.
• Work examples. Actually doing something for yourself is the best way to learn it, so take advantage of examples or exercises you can do as you read. Our textbook has such exercises sprinkled through the text.
• Ask questions! The math you’re reading isn’t supposed to be easy for anyone; if it were they wouldn’t need to write books about it. So if you get stuck, particularly if you come to a place where you can’t put an idea in your own words, imagine a picture or example of it, follow an example in the book, etc., then you have a question to ask. Ask it in an online discussion, an SI session, a class meeting, by email, one of our individual meetings, etc.

Do you have other tricks you use to understand math? What are they?

Try some examples of the above points:

A Basic Law

The book’s very first limit law is

How would you paraphrase this equation in words?

Can you give a concrete example of it?

Can you imagine, or draw, a picture that illustrates the idea? (I think you can upload small pictures in this discussion, or we can talk about pictures in class and I’ll post them to the daily class notes.)

Do you have any questions about this law?

A More Complicated Law

The book also gives a limit law that says

How would you paraphrase this equation?

Can you give examples of it?

Can you imagine or draw a picture to illustrate it?

Do you have questions about it?

Using Limit Laws

Try to use limit laws from the book to find the following limits. Also try to explain in a few sentences (or equations) how you used the laws (the process is more important than the final answer).