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202reviews - Math 202(Introduction to Real Analysis...

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Unformatted text preview: Math 202 (Introduction to Real Analysis) Calculus Review Exercises Notations: In these exercises, :1: takes on real values and 71. takes on positive integer values. 1. Draw the graph of the function f (cc) = g. 2. Find lim ESL—29933 m—>+oo 3-1—43; 2 ,. . , , 3- Letf(a:)={m lfm7é3.lsittruethatf’(x)={3m 1f$9é39 3:1: if a: = 3 if a: = 3 ' 4. Must 1°° = 1? More precisely, let a1, a2,a3, . .. be positive real numbers. Must it be true that if lim an 2 1, then lim a2 = 1 ? n—>+oo n—>+oo 5. We know that lim sina: doesn’t exist. If a1, a2, a3, . . . are positive real numbers with LII—V 00 k. lim (Ln = +00, then must it be true that lim sin an doesn’t exist? A "#00 n—->+oo \ '6. Show lim sinn aé 0. (By MCCV’CKES) W5 {5. {Kebacaum AW gin-X ’Ffldts n—>+oo {-5 M ‘to 94x??? I) 7. Let ( ) __ 1 if a: is a rational number in [0,1] g m — 0 if a: is an irrational number in [0,1] ' 1 1 2 n — 1" For every positive integer n, divide [0,1] into intervals [0, El’ [5, E] , . . ., [ n ,1]. ' - 1 ' . . . I On the j-th interval [‘7 ,1], let 233- be its midpoint. Slnce asj 1s ratlonal, we have n n 9(931') = 1- NOW ml ataxj areas «a? vacfmiles over +lu2 Mfewmls amt Tab/t) li‘mftj um gm; —0) +g<x2>(3 — 1) +---+g<x.)(1 — n‘ 1) = 1. n—>+oow_ W n n W n -—.l —— .— 1 S0 / g(a:) dz: = 1. Is this correct? 0 8. Let h(as) = {:2 sin(1/:r) 11: am: 7: 3. Graph h(a:). Find h’(a:). What is h”(0)? ...
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