2 LIMITS

CHAPTER 2LIMITSGROUP WORK 2: The Shape of Things to ComeThis exercise foreshadows concepts that will be discussed later, but can be introduced now. The idea is toshow the students that the concept of “limit” can get fairly subtle, and that care is needed. The second pageforeshadows Section 4.4, and the third page foreshadows Section 2.4. Pages 2 and 3 are independent of eachother; either or both can be used. Problem 3 on page 3 is a little tricky and can be omitted if desired.Answers:Page 11. 2 1 = 2, (0.6) 2 = 0.36, (0.8) 3 = 0.512, 0 4 = 0, (1.01) 8 = 1.08292.y2f§ f£föfªfÁ3. f n (1) = 1foralln.4. The curves all go through the origin.1_1 0 1 2 xPage 2_11. (a) 0 (b) 1 (c) 1 (d) 132(e)2. (a) 0 (b) 0 (c) 0 (d) 13. (a) 1 (b) 0The function f (x) = lim f n (x) is often seen in real analysis. It looks like this:n→∞y112560 1 xPage 311.102. Estimates will vary.√3. 10110− 1 ≈ 0.0049876. (For Problems 1 and 3, any number less than the given answer is correct.)4. Yes, the problems could have been done with any smaller positive number.44