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Techniques for Computing Limits

# Techniques for Computing Limits - \$1.4 TECHNIQUES 2r...

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Unformatted text preview: \$1.4 TECHNIQUES 2r; ConPUTINc LIUITS EXHnPLES: I. Im xzz 31=q X473 .1, lm cxlux) = c-nluL-u =-l Xé" Poumomms : M) = an x"+ + a,x 440 lm Pix) = Pm) ab»; = aha." +---+a,a-+Aa 2 I x -4): ~+ sans) -.-. 3. Im w 3 : ,3. 13,3 ’ --'9 3+, = Li H I z x—9x+3 -—-> t-‘iuiis =0 4. 1m T———-—- = 2— : 0 x—H x —2x-3 ——-, thaw-3 =-‘-I '"l 5. lm x-m x"— 1x -3 -——-W -H ——————’—'—_ SO Th’E LIDIT 0055 NOT EXIST xz-‘JXI'J ~—-? 0 mammal, FUNCTIONS: Pm) mm Gm) Pmywommg PL hr) X) -.~. Eff-)- UNLESS (MAL-:0 14’“ 00:) Q a.) IF QM) = 0, Bar Pm) #0) mi Lmn' 0053 nor 5x157. IF Gm) =0 AND Pm) = 0 THE Lmn' I5 \$310 To 8E. INDETERnwﬁTE gf TYPE 0 ﬂ 0 HM) mama‘- waRK IS REQUIRED To 05 TERmNE 'rmS LmIT' 5.6., nu. oF THE Fouowwc ﬁes INDETERDWAM" x3 x x3 5x3 [In 7 lm )(3 ’m 3"; [m :7 I 4x H m l : x—w“ x" [m JXH _ _w - x14 THIS TIDE THE "Ex 7M wolm ” REQUIRED TD DETErInwE THE LInIT LS PRET'ry oBVmUS -‘ FﬁCTM emo CRNCEL xz- ax-g cx-3)cx+:) [In 2 = ’10 x43 x "4" *3 x-vs LX-SJLX-I) X H = I» n ————- NOTE : camcstuvc WE x-vz 1" x-g nEANS DWIDJNG Nunﬁknmk BNO DENommm-og BY X~3 . THIS 15 k5 cmnme BEcnusEJ "0 hn X#3, x-as ’ 3 I _ = 32'? ‘ 4 PM ——»0 Im #41 do 1-4-1 . WI 35cm.“ = n tie-z ’H'L 33+ 33: (B+B>(Hl-ﬁBiBz) 33- :33: (19-13) m‘mn +81 ) lm (kl-1* W) aha-z § - L..7_)L-:.(-L) + H = u a .- (1+h)3-9 —a o 9, lm -———-—- 5+0 h -+ 0 THIS TIDE. :1 IS PﬂoBﬁBLy 505/657“ 7'0 EXPﬁND 3 (MHz-9 C8+I~25+6l12+b )-3 Inn ’1 = Im ————--——‘—'""‘_‘h baa hqo tab 4451+“ Fukuoka!) : hn “——:———‘ = hn -—-—:———- ham 51-90 = Im (n 4011”?) = [.14 4.0 40‘ =12 hqo «I'X " *‘5 0 1a hn ‘—jf" _% o x—w TIfE STﬁNDﬁRD THING TD DO WNW I9 SQUHIPE ROOT IN [9 sun on DIFFERENCE IS W r7): - 1" l p—x -1 l ﬁ-‘x -I m: H In C” on = m -—-"""'"' = l XLW-x H) xao x xwo x W'X” x40 l-X -I '7‘ l -’ _ Im = Inn —-r—_::-"‘;‘ = m X40 XHI-x H) 30.90 X( I“! H xqo Vl-X 4” -! : .———-——-——" z .- ‘L & H, H .._L_. _ .L. hn “‘1' a Jr-vo .é ETC-I. " 1" lm : ,b-ao 16 (m .2- anti) ___._.’ I Im an ~31— 26-90 1: L -t = Im aunt) M 2(2411) " t —-3 o h *“7' = g = o I m 1 X ‘ “-2 FCX) : {as ,-.24X£3 ““3 x > 3 I hn FLX) = Im 17;: : -oo Xa-z‘ x—a-z 7. Im Rx) = Im 0&5) -. (-z) -5 z -1 x-a-z‘ x—a-z" 'L , 7:. ‘, [In pcx) lln (XI—5’) : o “b .5 x40 xao 7. lm PM) Im' LXI-5) .-. 3 —5=L/ THUS) -> ' 31-93 x 3 .1.’ 1m pLX) lm [3m - 'm + 41413 = ¢3+I3 * 7H3 ...
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Techniques for Computing Limits - \$1.4 TECHNIQUES 2r...

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