2 LIMITS

CHAPTER 2LIMITSGROUP WORK 2: The Dire Wolf Collects His DueThe students will not be able to do Problem 1 with any sort of accuracy. Let them discover for themselveshow deceptively difficult it is, and then tell them that they should do the best that they can to show what ishappening as x goes to zero. Ask them to compare their result with lim x sin (π/x). If a group finishes early,x→0pass out the supplementary problems.Answers:1.y1_1 1 2 3 x_12. (a) 1, 1, 1 (b) 1 (c) 0 (d) A function must approach only one number for the limit to exist.Answers to Supplementary Problems:1. The length of the boundary is infinite. There are infinitely many wiggles, each adding at least 2 to thetotal perimeter length.2. The area is finite. It is less than the area of the rectangle defined by 0 ≤ x ≤ 1, −2 ≤ y ≤ 1.3. Answers will vary.GROUP WORK 3: Infinity is Very BigThe precise definition of infinite limits is similar to the standard definition, but it is different enough that moststudents need a little practice before they can grasp it.Answers:1. x < 0.0012. (a) Choose M. Nowletδ = 1 √M.If0< x < 1 √M,then 1 x 2 > M. Valuesofδ less than 1√Mwork,too.(b) 1 xis large negative for small negative values of x, and large positive for small positive values of x.GROUP WORK 4: The Significance of the “For Every”The purpose of this exercise is to allow the students to discover that rigor in mathematics is often necessaryand useful. Problem 1 is designed to lead the students to make a false assumption about the third function,h (x). Problem 2 should dispel that assumption.60