What did we do?
We discussed about limits and continuity, but in a different order as in the book, although we talked about the same topics. We motivated the idea of limit with the example of understanding the number pi and how we get approximations to it.
We discussed mostly continuous functions and their properties, in particular the intermediate value theorem and how to use it to find roots of polynomials. Finally we discussed the concept of limit and how can they fail to exist by explaining left and right limits.
What we did not do?
We did not discussed limits and asymptotes in length.
We discussed how limits and continuity can be preserved by operations as addition, multiplication, composition, etc, but we did not write the properties explicitly. These appear as theorem 1.2, 1.3 and 1.4.
We discussed it in tophat problems, but we did not discussed it at length the limits at infinity and the squeeze theorem.
Comments on section 1.7.
- At the end of page 58 it explains that the idea of continuity is important because small errors in the independent variable lead to small error in the output. This is, in essence, correct but it is important to emphasize that how small the error in the input should be to guarantee a small enough error in the outputs depends on the function. The same mistake in different functions can lead to very different mistakes in their values.
- We did not discuss an example similar to Example 5, so it is important to read this carefully.
Comments on section 1.8.
- Example 2 is important to understand. The reason for why they are important is because they are the “simple” type of jumps that destroy cntinuity, we call them jump singularity, and relate to the fact that left and right limit exist, but go to different values.
- Example 4 is important to understand. It relates to the previous example, when the left and right limit are not numbers but infinities.
Comments on section 1.9.
- It is very important to remember the cases to evaluate quotients as explained at the beginning of this section (the three dots list).
- Theorem 1.5 is important. We discussed it related to a problem of tophat. In particular, to have the figures in page 79 in mind to remember this theorem and how it works is a good idea.