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# Class Note for MATH 1431 at UH 3

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This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Lecture 2Section 22 De nition of Limit Section 23 Some Limit Theorems Jiwen He 1 Review 11 The Ideal 0f Limit Graphical Introduction to Limit o In taking the limit of a function f as x approaches c7 it does not matter Whether f is de ned at c and7 if so7 hoW it is de ned there 0 The only thing that matters is the Values that f takes on When x is near c 12 Examples 2 7 9 Example lim x was x 7 3 Let 279 Use the graph of f to nd 0073 af3 3 lim 1 C 11m 1 d1imfw zHB z 3 1H3 Use the graph of f to nd af5 3 lim 1 C 11m 1 d1imfw zHS z 5 1H5 Example Use the graph of f to nd af3 3 11111 x C 11m x d 11m x magi 1H3 ef7 1 lim 1 g lim 1 h1imfw z 7 z 7 1H7 7V Example hm sm 7 3H0 z 139 Emit Let sin Use the graph of f to nd a MD b 111 NC C 11m x d 11m x OHM 3H0 13 Theorem An Important Theorem Theorem 1 hm L and only both hm L and hm L z c an c 2 Section 22 De nition of Limit 21 De nition of Limit Re nitg n of Limit 5 6 statement 6 say at hm fz L if for each 5 gt 07 there exists a 6 3 6 such that if 0lt zic lt67 then lt5 L r C 5 C C 5 x y y y y L L 23 L L fx I 1 O x I 3 I C x I C x I C I x I x C A It 4 If t I For each 6 gt 0 there exists 5 gt 0 such that if 0 lt Ix cllt 5 then Ifx M LI lt e I Choice of 6 Depending on the Choice of e y LE L1 L E y fx2 f L Ifquot I l L quot T 39 fX1 4 i l I I I c 5x1 c x205 x Example y I L 6 r Which of the 6 s works for the given 6 a 51 b 52 C 53 Example gtltv For Which of the 57s given does the speci ed 6 Work a 51 b 52 C 63 22 Examples Example lim2x 7 1 3 Show that 11fo 7 1 3 Let 5 gt 0 We choose 6 5 such that if0lt x72 lt6 then 217173 lt5 Example lim 2 7 3x 5 Show that 7 3x 5 Let 5 gt 0 We choose 6 5 such that if Olt xi71 lt6 then 273x75 lt5 23 Four Basic Limits Basic Limit lim x c Show that limwf E c was 2 fx x Let 5 gt 0 We choose 6 5 such that if0lt acic lt67 then xic lt5 Basic Limit lim c Show that 11111 sz gtltY IIm Ixlc XC Lets gt 0 We choose65 such that if Olt acic lt67 then Hz i cHlt5 Basic Limit lim k k Show that linfk E k was kEi39 39i In I I fIxIk I I hequot quotI I I I I gt 2 5 C 65 X Hmkzk X c Let 5 gt 0 We can choose any number 6 gt 0 such that if0lt acic lt67 then kik lt5 Basic Limit lim E Show that for zc 07 11111 V9 E was 10 2A fxt lim2 xa4 Let 6 gt 0 We choose 6 minc e such that if0ltixacilt6 then fa lte 3 Section 23 Some Limit Theorems 31 Some Limit Theorems Some Limit Theorems Sum and Product Theorem 2 If hm and hm 933 each exists then 1 hmltfltxgt m gig f33 53 gm mac 2 hma a hm mac mac 3 hm hm 933 mac mac mac rggr iinngif T r hngliiiiiiggst each exists then hm 1 zf 7E 0 then a hm mac 11 2 if lim 91 0 while lim f 0 then lim does not e1ist mac mac mac 9 1 I 5 if lim 91 0 and lim 0 then lim L may or may not e1ist mac mac mac 91 Limit of a Polynomial Theorem 4 Let P1 an1 a11 a0 be a polynomial and c be any number Then lim P1 Pc E1amples 5 lim1213 12 a 21 a 3 2a13 a12 a 2a1a 3 a2 lim1415 a 712 21 8 1405 a 702 20 8 8 1a0 Limit of a Rational Function 131 QI nomials and let 5 be a number Then Theorem 6 Let R1 be a rational function a quotient of two poly 7 im P1 Pc C 1 if f 0 then R1 a 40 R 2 if Qc 0 while 135 a 0 then gig Rltzgt gig 8 does not e1ist 5 if 0 while Pc 0 then R1 may or may not e1ist Examples E1amples 7 3 7 2 3 7 2 hm 1 31 7 3 33 7 27 27 7 0 was 1am 143 1a9 2 2 2 does not emst7 hm 700 hm 1a1171 1a1 1 1 ma1171 127176 m17312 mas ma mas ma ilhm 71 1hm 1 1 mai 171 ma1ma1E1 ma1 1 2 007 lin12325 12

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