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2011m-solution

2011m-solution - MATH 117 Midterm 1 Winter 2011 Feb 9 2011...

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Unformatted text preview: MATH 117 Midterm # 1 Winter 2011 Feb. 9, 2011 Instructor: G. Wei San/MW; Name All questions have equal points. Show complete work. Continue on the back of the page if you need more space. Good luck! 1. Show by deﬁnition that the sequence {ﬁﬁil converges to 0. We “we? t 54m V629}_ saw mg i; ,ﬁ/fé W1; MM 5” i. _...—— ,. ___ i _ , [ﬁr 0/ ("i 4-) 7;: 4’5 4-9 > _§€_ w u, > ___... New (L? N:[_ZL:J+I WK“ M24} 5 I r i? > 35:- {JL— we & 2: z‘" WT 2. Deﬁnef:[—3,0]—>Rby f(\$)=2£2:3§_2form7é—2 and f(—2)=—5. Isf continuous at —2? Justify your conclusion. M .. "w? 1w x) F! f' 1;,sz ‘ ‘2 A '~ % n—- .. - ETHVZ dim ; 2X 3X 2 5 [2X i)(«\"+.&j f gym ﬁx”) :3 {m (2%)) ; air We»), )(e-Z ‘ 9’ / i .' _ . .u I’m 7L”) " 7W4)! f ’4 5°”“W a»? 02, 3. a). Let S be a set of real numbers and a = sup 5'. Show that, for any 6 > 0, there is a point m E S such that a — E <m S a. b). Show that there is a sequence {anﬁil from S converging to a. £1)(?mﬁaﬁé[WW£Mi¢f/1c’yf 2’70 avg «:5 M M: WW I" fie-5' me; if v m p are; Wm mé‘fﬁ m £01: //Wee%-{é’m§q 191"“ 571'“;ch Mammy {thp-aqi 4. a). For a function f : D —> R, and 230 an accumulation point, give the deﬁnition of lirnmwwcO = L. b). Show by deﬁnition that 11inan :52 = 4. i3) (11%.?0) 9 (9’70 L‘f' o£[3(",\’g( (5’ K99}, I [ffx'JILf (f a.) (X1,ch ma; [x-vz/ WA?» [X‘fzféf’ (£144.. 5. a). Give the statement of Bolzano—Weierstrass theorem. b). Show that a. bounded sequence has a convergent subsequence. (Sorry, only quoting a homework is not enough. Please Show your work.) g“) My éwM vhf?”wa w 4 mf mg“;- 44,; cu [emf woe mwwme FM_ “1% ,QMtée/mgam §aL: as :a whey/kw? “ﬂag :5 deaf/#M. {if ﬂthéfjxyi‘ Cma‘; {at} to “we WWW M a. M, my Bag/333w a r Aer/5 (gm awemwwﬂ PM, W A {a} Mute a; Awfwﬁﬁn 1L“, V570 I‘ WZ/Aéwicmﬂf [ﬂ~£/ Mi) [wife/{‘51 “£hl’6; mu? [xx-W‘s WWI [ex 677%. JW 53% 7AM té ﬂaw, mi ﬁﬁwé[4%,mak) 11\$ mpﬂw. ﬂy,“ Mn“? 13! it mé7w¢ W? ﬁA, 3 ...
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