MATH 301 Week 5 Homework, Fall, 2016. Due Thursday, October 20. NAME: _____________________________________ Instructions: You are encouraged to discuss homework, use online resources, and to seek help from the instructor when you need it, but your submitted write-up of your work must be your own, in your own words. There are 8 problems, most with multiple parts. D is often used to denote the domain of a function. NOTE: We cannot apply L'Hopital's Rule since we have not even learned about derivatives yet. LIMITS #1. Determine each limit and prove it, using the - definition (LImits Notes, p.1 or Def. 3.1.3 in Lebl). #1(a). lim\u0004 \u0007 (5 \u0007 12) #1 (b). HINT: See Example on page 2 of Limits notes \u0012 lim\u0004\u000f sin \u0004 Page 1 of 6 #2. Find the following limits (if they exist). (You can apply results and theorems in notes and in Lebl; do not need to work with the definition of limit) Explain/ show work. If a limit does not exist, explain why. It can be helpful to look at pages 4-6 of my Limits notes. [Note that we have not yet covered differentiation, and thus cannot apply L'Hopital's Rule.] \u0004 \u0014 \u0015 \u0016\u0004 \u0015 \u0012\u0017 #2 (a). lim\u0004 \u0013 #2 (b). lim\u0004 \u0015\u0007 #2 (c). lim\u0004\u000f cos \u001b \u0014 \u001c \u0004 \u0004 \u0014 \u0018 \u0013\u0004 \u0018 \u0007 \u0004 \u0014 \u0015 \u0016\u0004 \u0015 \u0012\u0017 \u0004 \u0014 \u0018 \u0013\u0004 \u0018 \u0007 \u0012 Page 2 of 6 Continuity #3. Decide if each statement is True or False. If False, provide a counterexample (specifying D and the functions involved. No proof of true statement requested. Explanation of counterexample not required. ) #3 (a.) ________ If f is continuous on D, then |f| is continuous on D. #3 (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.) Define \u001d( ) = lim\u001f (\u0012 \u0015 \u0004)! \u0012 \u0018 (\u0012 \u0015 \u0004)! for all real numbers x 2. (a) To get a feel for how the function behaves, determine each of the following values. That is, substitute a given x-value, simplify as appropriate, and then find the limit as n . f (2) = _____ f (1) = _____ f (0) = _____ f (1/4) = _____ f (1) = _____ f (3/2) = _____ 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. Just state your formula. (c) For what values of x is f continuous? (no explanation required) Page 3 of 6 #5. Let f: D R be continuous. Decide if each statement is True or False. If False, provide a counterexample (specifying D and f; note that your function f must be continuous.) (No proof of true statement requested. Explanation of counterexample not required. ) #5 (a) ________ If D is closed, then f(D) is closed. #5 (b) ________ If D is bounded, then f(D) is bounded. #5 (c) ________ If D is compact, then f(D) is compact. (Recall that compact = closed and bounded.) #6. Prove that 2x = 5 cos(4x) for some x in the interval (/4, /2). (You can assume the cosine function is continuous.) HINT: Let f(x) = 2x 5 cos(4x), and show that f(x) = 0 for some x in (/4, /2). Verify in writing how the Intermediate Value Theorem can be applied to f on the interval [/4, /2] to produce the desired result. Page 4 of 6 Uniform Continuity #7. Each of the following functions f is continuous on the given set D. Determine whether f is uniformly continuous on D. Justify your answer by providing a brief explanation (just a sentence or two, no formal proofs). HINTS: It is helpful to go through the *Theorem and example on page 10 of the LimitsContinuity notes. #7(a) \u001d( ) = #7(b) \u001d( ) = #7 (c) \u001d( ) = \u0016" D = [1/5, 6] \u0004 \u0016" D = (0, 6] \u0004 \u0007\u0004 \u0014 \u0015 \u0016\u000f \u0004\u0015\u0016 #7 (d) \u001d( ) = sin 1 D = (0, 5) D = (0, #) Page 5 of 6 Limits at Infinity and Infinite Limits #8. Let f: (0, ) R. The goal is to use the limit definitions in section 3.5 (Lebl) to prove: If lim\u0004 \u001d( ) = , then lim\u0004 \u0012 $(\u0004) = 0. Fill in the blanks below. Each __ can be filled in with < or >, as appropriate. Discussion: Referring to Def. 3.5.1, suppose > 0. \u0012 We want to find real number M such that & 0& __ whenever x M. $(\u0004) This means that we want \u0012 |$(\u0004)| \u0012 __ or equivalently |\u001d( )| __ . Applying Def. 3.5.6 (Lebl): Since lim\u0004 \u001d( ) = , \u0012 given ( = , there must exist a real number M such that \u001d( ) __ ( = Note that for x M, we must have |\u001d( )| = \u001d( ) since \u001d( ) __ \u0012 \u0012 whenever x M. \u0012 and __ 0. Proof: Let f: (0, ) R. Suppose lim\u0004 \u001d( ) = . \u0012 Let > 0. Set ( = . Since lim\u0004 \u001d( ) = , by Def. 3.5.6 (Lebl), there exists real number M such that |\u001d( )| = \u001d( ) __ ( = \u0012 whenever x M. Thus for any x M, & \u0012 $(\u0004) \u0012 0& = |$(\u0004)| = \u0012 $(\u0004) __ \u0012 ) \u001b\u001c = *. Hence Definition 3.5.1(Lebl) holds and lim\u0004 \u0012 $(\u0004) = 0. Page 6 of 6