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# Class Note for MATH 1432 at UH

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COURSE
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KARMA
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This 7 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 16 views.

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Date Created: 02/06/15
Lecture lgsection 105 Indeterminate Form 00 Section 106 Other Indeterminate Forms 00007 0 oo7 Jiwen He 1 Indeterminate Forms 11 Indeterminate Form 00 WEIMalia 1the Indeterminate Form 00 1 17271 172 71 1 7 1 71 1131212747113121721271g1272274 1 1 1 L Hapital s Rule 1 Amazing7 As1gt27 f1172gt0and911274gtO7 but the quotient 7 172 HO 91 71274 0 We can not determine a value for g quot without some e1tra work If lim lim 91 07 then lim might equal any number or even fail Ha Ha Ha 91 to exist 1ex Condensed The arm is indeterminate 12 Other Indeterminate Forms Indeterminate Forms Indeterminate Forms 0 The most basic indeterminate form is g 1 o It is indeterminate because7 if lim lim 91 07 then lim amp 1 H H 91 might equal any number or even fail to exist sin1 e o SpeCI c cases 11m 7 11m 7 1 Ho 1 H00 sumI o The forms and 0 00 are also indeterminate 0 They are actually equivalent to since any speci c case of one can be recast as a speci c case of another 0 For example7 1 1 7 1 lnz 7 z 133 I 7133 11 7133 11111 00 0 0 i 7 00 00 0 Quiz Quiz 1 Give the limit for l 7 n 711 n2 a 07 b 17 c 2 2 Give the limit for 1 n n2 a 07 b 17 c 2 711 2 L Hopital s Rule 21 L7H6pital7s Rule L Hopital s Rule L Hopital s Rule Suppose that lim is a 9 or El Then lim amp lim Hgm o co 17m Hyman Examples 2 170 z 0 70 1 z 7 sinz 0 l 7 cos I 0 hm 7 hm 7 still 170 zslnz 0 x70 s1nzzcosz 0 sinz 0 m i 0 70 coszcosz7xs1nz 2 What is Wrong with This What is wrong W12th the following argument z 0 2x 2 2 hm 7 hm hm 700 xao smz 0 x7gtO cosz x7gtO 7 smz 70 Remark LlHopitalls rule does not apply in cases Where the numerator or the denominator has a nite nonzero limit For example7 2x 0 1m 7 07 xao cosz l but a blind application of L7Hopitalls rule leads incorrectly to 21 2 2 lim 1m 7 700 xao cosz x7gtO 7 smz 70 Quiz Quiz 00 3 Give the limit for a 17 b 27 C 6 4 Give the limit for lt1 a 17 b 27 C e 7 711 3 More Examples 31 Form Ewumps 3 sinz7z 0 cosz7l 0 7 7 7 7 13 3 lt0 13 3352 lt0 still lim 31 still lim 70 i 1H0 6 6 7 2 7 39 lim2cosz 21 lim 2s1nz2z still 1H0 14 0 1H0 4x3 72 cosz 2 0 7 39 l 0113 12x2 lt0 stzll 711mm 9 still7limgcosx737i wo 24x 0 quotwe 24 724712 1m ln3 2 71 311n2211n2 imam 31 lim EH00 I 00 ln3 23 an 7 ln30 1 7 1 3 11330 123 10 n 33 Limit Properties of lnx and em Limit Properties of lnx lim zalnz 07 a gt 0 zao Proof lim zalnz lim mi 0200 xHOi 1 oo 1 a 11m 79 iim L0 0200 iaz a l xa0 7a lnz hm 7 07 a gt 0 EH00 Ia Proof limlnizltggt lm 11 Him 1 H0 EH00 Ia 00 EH00 110471 EH00 110471 Remark 0 lnz tends to in nity slower than any positive power of It 0 Any positive power of lnz tends to in nity slower than 1 Limit Properties of e lim i 0 THOO ex Proof In 00 nznil 00 hm i hm THOO ex 00 THOO ex 00 n n 71 zn 2 00 1m i mace e7 00 l l n n hm i i 0 THOO ex 00 Remark 6 tends to in nity faster than any positive power of 1 34 Exponential Forms 1 00 and 000 Exponential Forms 1 quot7 007 and 000 Since ln 117 plnz7 the log can be used to convert each of exponential forms 1 quot7 007 and 000 to 0 00 lnl 00 07 ln00 00 07 ln000 0 00 If lim lnfz L7 then lim eLi 1 mavt Example 5 lim 11 00 60 l EHO l limJr lnzx ln 00 Thigh zlnz 0 00 EHO lnz 00 lI 11331 a 12 T12 EEIBA IFO Emm x onential Forms 1 quot7 007 and 00 lim 111 000 60 l mace 1x oo 7 1 Egmlz l e e lirn lnzlx lnooo 11m mi hm 1710 700 1700 x 00 700 1 lnlz 0 1 1 1 10 1 1 1 7 7 1111 nlt W n gt 111 I 0 lirn 1 1 xaolz 0 Set I n7 we have the familiar result as n 7gt 007 mi 7 l1 0 Setzlnwehave asn7gt007 1iln le 35 Form 00 7 00 Form 00 7 00 The form 00 7 00 can be handled by converting it to a quotient1 39 71 0 lirn tanz 7 seem 00 7 00 lirn amp lt7 177r2 177r2 cosz 0 7 lirn CO 7370 7 177r2 7sinz 7 71 7 7 equot 71 0 lirn Eagle 761 00700 lirn 67 lt7 1700 1700 5 1 0 equot 7 7x it lirn w lirn ee eol EH00 767 EH00 sinz l sinz 71 tanz 7 seem 7 7 cosz cosz cosz exleiz 7 ex 616871 71 76871 71 671 Out line Contents 1 Indeterminate Forms 111 00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 112 and Others 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 L Hopital s Rule 211 Rule 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 3 Examples 31 g 32 g 3 3 lnz and e 3 4 Exponential Forms 3 5 00 7 00

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