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Techniques for Computing Limits

Techniques for Computing Limits - TECHNIQUES 5:313 ConPurmc...

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Unformatted text preview: TECHNIQUES 5:313 ConPurmc EXflnPLES : I, [m X1: 3"“3 X—‘r3 Lmn‘s a. lm (xhax) = {-1)1+Ql-I)=-l 'X-fil-l Pouwomms : lm X—Ma. Pix) = anxn+ ---+a.x 440 Pm = Pm.) n = “If!“ +~“+Q‘a-+qa xz-ax —-+ 32— .us) = 3 3. Im an : .3. 15—93 ' “’1 3+I -'- H a z z , X-HJH3 --+ :-%J+3.=o a ‘L 'I” 7““ 1 = —— = a 1—H x--.?x~3 ...., i-au)-3 s-..) .q x1— .17: ~3 -—-» #1 5. Im ._.._.....—--— a so THE leT 0055' Mar EXIST 3197:0an FUNCTIONS : Fax) nun 0w) PaLynomeLs_ Pu) P I!” ) '.-. 4‘21!- UNLESS aid) :0 2-54. an: 6? 4) IF QM) = 0,1301 Pm.) #0, THE anr 0055 N07 EXIST. IF GM.) =0 am; PM.) = 0 THE L101?” 15 53:0 To BE woarerenwmf 9f TVPE O .— 0 ant) 7108!? we“ IS REQUIRED To DE JEWINE TmE umr 5.8., an 0F THE Fouowwc fies :moETERmnMJE ,m X’ , .36.. , .53 I 5X’ _:k—- . In 3 H1 3 "1 3 I “‘3' 4“) H = (m M4? 1 3 so me Lmn ONE. 39.4,, Xvi —--Ip 1—! = 0 l 42): H n .-. )c-l-u" x1—: I,” 43:?! = ’69 - x14 7.‘ 1m THIS was 7717? “EUR/9 may " REQUIRED To DErErinwE THE erT '5 PRETTy oBVmUS : I FACTOR awn CHNCEL [m xz—ax-z I n = I X-‘93 Xqu *3 3:493 = 1m 1-33 . 3+ - “—3-; £3+9 I —9 O 8' £31,”; £+1 do I {final—MM) = In M H”; but :2 III? (tl'u*q) .t-w- —_ c—n'zu—uw = u. cx-3)cx H J Lx-szcx-U NOTE : camcELst ms x—3 nEAms owwwc munERBTOR 3ND DENanmgmg 3y X-3. THIS IS kEcmnm'E 3565055) no rm x¢3' x-pa BECBLL : 19533 33-83 = (HENRI-331F313 .-. {19-5) m‘HJB +32 )- (Jung-9 ‘ —-> o 9 'm -"'_——“"' ham h a a T815 "runs :1 15 Woman 5851557“ 7'0 EXPrmD ' 3 mung-9 (8+l-117461114L’7J-8 hn “37—" = 3m W5 have 5‘99 . 1314053“? “’4405451) : hr: ““7"” = 1'" I, h—ao ' (“'30 .-. lm unwind") =. 1146-040?" :12 h-flo I'X " "‘5’ O 10. [In x a 0 1+0 THE STfiWDfl/c’D THING TE? DO WITH [9 SQUHEE ROOT IN I? SUI“) OR DIFFERENCE I; WE ' 1 2. - -l l 1—): -J ' Iii-x -I JI-x H In (F?) n = m ---—'—"'" = a thr H) 3:190 1‘ Xe» 3‘ W“! H x-aa x l-x -I " l " _ [m ‘— [In M ‘= m .__.....—— " Tum—H) x—yo XUVV H). x-ao “-3! 4" X-‘JO - l .__—--'——' = .- _L {3-0 H ’~ ‘I H. Ll. [In $.30 1" --*.~ 0_ w E ConPLEx FRHCTIOAJ ' -—-'- .. J. I -..L 1. _lm ‘1“ " 9- : [m am”; i um) I?” *‘w 1: aczm‘) . (m J-cau‘) I ~46?“ .. ' m 16—90 Mani?) PM .21; an?) ' 1m 6.2.5..— ‘9' 0 6 AH, aunt) ._.,,, - ”J = 0 ,3... x“; F at”, , (. : X) “1'5 , n1 2- x s: 3 ”“3 x > 3 I hn_ch)= Im_:;;3:_oo ’9’“?- x—‘a-z 7. lm PLX) =~ lm (X135) '-'- (-1) -5 = -’ “*‘f 1-3-2 1. ' _ 7- , 7’ _5. lm Pm) - Im (at —5') _ o as x40 xaa z 1 __ lm pa) = Inn- (x ~5)=.3-5,q THUS) X-‘IS' ”1-93 pt [In X)="I '10 Put) .- lm *1/1413 = u/3-ug .6 1+3 x-os _ ...
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