6.3.5 Study Project: Satisfying requirements on limits
Study Project: Satisfying requirements on limits In this study, it will be helpful to refer to the definitions of limits constructed in Discovery Exercise: Other types of limits of functions. 1. Functions with a limit on only one side a) In previous work, we have seen that a function that is not defined on values greater than p cannot have a right-­‐hand limit at p. Use this observation to construct a function that has a left-­‐
hand limit at 5 but not a right-­‐hand limit at 5. b) Suppose we now wish to consider functions defined on an open bounded interval around 5. Find an unbounded function f : (0,10) →  that has a left-­‐hand limit but not a right-­‐hand limit. c) If possible, find a bounded function g : (0,10) →  that has a left-­‐hand limit but not a right-­‐
hand limit. d) Which of the functions in your examples above has a limit at 5? If one of your examples does not have a limit at 5, can it be modified to have a limit at 5 and still satisfy the required conditions? Explain your conclusions. 2. Unbounded functions that do not tend to ∞ or −∞ a) If possible, find an unbounded function h :[−3, 3]→  that does not tend to ∞ or − ∞ at any point in the domain, neither from the right nor from the left. b) Your function in Part (a) may have involved a definition in two parts, for values of x coming from different subsets of the domain. Fill in the blank below, using one formula for all nonzero values of x in the domain: The function k :[−3,3]→  such that k(x) = __________________________ ∀x ∈[−3,3]− {0} and k(0) = 0 , is unbounded and does not tend to ∞ or − ∞ at any point in the domain, neither from the right nor from the left. c) Suppose f :  →  satisfies lim f (x) = 0 and lim f (x) = 0 and does not tend to infinity at x→ ∞
x→ −∞
any real number, either from the right or from the left. Does it follow that f is bounded? Either explain why this is true or give a counterexample. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810