#1. Decide if each statement is True or False. If False, provide a counterexample (stating domain D and the functions f and |f| involved; no explanation required). No proof of true statement requested. (a.) _ _____ If f is continuous on D, then |f| is continuous on D. (b) _ _____ If |f| is continuous on D, then f is continuous on D. #4. For all parts, just state your answers; no explanation required. (You will need to do work to determine the answers, but you are not required to show it.) ( x+1)n n Define f ( x )=lim n 1+(x +1) for all real numbers x 2. (a) To get a feel for how the function behaves, determine each of the following numerical values. That is, substitute a given x-value, simplify as appropriate, and then find the limit as n . f (1) = _____ f (1/8) = _____ f (5/4) = _____ f (3/2) = _____ f (0) = _____ f (5/2) = _____ f (1/2) = _____ f (2) = _____ f (3) = _____ f ( 4) = _____ (b) Determine the numerical value of f(x) for each real number x 2 ----- you should find a relatively simple multi-part formula for f, with domain {x | x 2}. Just state your formula. HINT: It can be helpful to plot the points you determined in part (a) to get a sense of the pattern. (c) For what values of x is f continuous? (no explanation required) Page 1 of 3 Page 2 of 3 #5. Let f: D R be continuous. Decide if each statement is True or False. If False, provide a counterexample, stating D , f and f(D); no explanation required. Note that your function f must be continuous. No proof of true statement requested. (a) ________ If D is bounded, then f(D) is closed. (b) ________ If D is closed, then f(D) is bounded. (c) _________ If D is compact, then f(D) is compact. (Recall that compact = closed and bounded.) Page 3 of 3