SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
This discussion should help you understand the Squeeze Theorem and its use to find limits involving sine and cosine that are, ultimately, needed in order to find their derivatives. All of this material is presented in “The Squeeze Theorem” in section 2.3 of our textbook.
Post at least one response to at least one of the following by class time Friday (October 2):
Suppose f(x) and g(x) are two functions that are defined everywhere near x = 1 (but not necessarily at x = 1 itself), and that wherever they are defined, f(x) ≥ g(x). Furthermore, suppose
\[\lim_{x \to 1} f(x) = \lim_{x \to 1} g(x) = 17\]How could you use the Squeeze Theorem to show that the average of f(x) and g(x) also approaches 17 as x approaches 1, i.e., that
\[\lim_{x \to 1} \frac{f(x)+g(x)}{2} = 17\]The part of the reading that derives values for limits of sin x, sin x / x, etc. is probably some of the more difficult reading you’ve done so far for this course. This part of the discussion tries to draw attention to some of the key ideas and common themes in the reading — hopefully, thinking about those things will help you follow the reading, or at least get its overall sense. You might find it useful to flip back and forth between looking at (and joining in) this discussion and doing parts of the reading, so that ideas from the discussion can help you with the reading, and questions about the reading can give you things to look for or say in the discussion.
What are the limits that are calculated in this reading?
How does the book use the Squeeze Theorem to derive some of those limits? Is there any common pattern, theme, or strategy to how the book uses the theorem?
What do the diagrams of unit circles (Figures 2.29 and 2.30) contribute to the derivations?
The book derives some limits without directly using the Squeeze Theorem. Which limits are they? What method(s) does the book use when it’s not using the Squeeze Theorem?
What questions do you have about the reading? (Those questions are things I can discuss in Friday’s class, or that others reading the discussion might be able to answer.)