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## Calculus 1 week 1 notes

by: Sarahmariponce

5

1

14

# Calculus 1 week 1 notes Cal 2314

Sarahmariponce

GPA 3.6

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Limits
COURSE
Calculus 1
PROF.
Dr stalanski
TYPE
Class Notes
PAGES
14
WORDS
CONCEPTS
Calculus Limits
KARMA
25 ?

## Popular in Natural Sciences and Mathematics

This 14 page Class Notes was uploaded by Sarahmariponce on Monday September 5, 2016. The Class Notes belongs to Cal 2314 at Tarrant County College District taught by Dr stalanski in Fall 2016. Since its upload, it has received 5 views. For similar materials see Calculus 1 in Natural Sciences and Mathematics at Tarrant County College District.

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Date Created: 09/05/16
Limits ibmtinuih lk One lrlimitsand . continuity " limit notation )=L thblimitatftxlasx Approaches C " ×→c ° fklisaosbtolfx XCAN Never be Atc ' " Dhtitcanbbadsb Karan . usingwmitswelananalyzeabenavidrotafunaimasweapprdamavalne in its Domain . howdowedothis ? -Usb the three For AHATCAIANAMOAT problem solving Strategies . Dnumerically aenstrwtatablb - oharapnically drawaarapnoruse graphing calculator . ehanamtically usingalaebraoroalcuvus . 4) Formal Prootlprat added) - ASKYMSHF .. howdowlappnlachthl domain ? fromthbltfti . from the right . smaller man =/ - vavqerman input input input . Exatfindingalimit numerically . ) -3×+2 findthetimiX→2 g ,.q 1.99 1.999 limfk .at#t2Hd::ki8kk.kkktAsWbapprdach2frMthe1eftandtwofrMtherightftX )→8 note : limmustbethesameaswbapproachfrmtheleftandthbriqhtltwosided ) Exiuseatabktofindxhjmfktxxtztxtt sofindvawlesot-aosetol . ' X 1.999' 2 -2.0112001 1.99 FIX )=X2# 3.993.999 4.014.001×-2 v im=4 Exotfindinlalimitgraphically .• him FLIXII x→2 ritirkn 2 Exigivlnmeqrapnfindthqyjmofk ) axsx DN{ ASX→o fromthe left fk )= -1 ,×T0frommbri9ht}µm Notlitfwboblnihadifferentvalubaswbapproachofromthblbftahdthb Viahtthbnlimfk ) DdExistt . x→c EX : limflx) limfklfrmleft :approach , " ×→% y\ : } M= 2 limftxltromrrqht approach , remember : alimitthataumntsforbom approaching frrmthbleftandviqht isatwosidedlimit . 1) ( limfromltfttx Ytngtromrigntafamesalytmftxka * '=0 asymptote toqetaneaohhmsfks Prootruleimath Language limfk )=L means ... ×→ctq>o then - ,Faf>olif/xt/(8 , IHH YCE f: form I : thereexist 1 whatddesthismban ? : Aletta Forauepsilonyotheroexistadeltalosucn - §: that ifthevalneotxtcothenfk ) L< E . :MpYiYhAt}4 c- fcxtcg Scxlcts -E ( flx)-L ( E L . E ( fk )( HE ⇐ Epsilon tubb Y th L eHthblimitlsliithastobeinthb ←tsldcts → tubhinmb interva. interval ProofexplainedAMDHS • for theres we : asxgetsmiaaehwblmvlrqb a any positisuch that oraetaoserwl positives, :: . •. . )=L Xisgreaterthanfthen : hmfkx the distancebetween Mr 23 4 56 Pdintsandthblimitisless s oxwnvergesioi. than E. aslmgasmrxisgreaterthansa ) then Weave Within EOFL , EX : When 4=0.3 and 0.006 prove hi→mfW=3Xt2=8 finds FEIOIAS Calif 1×-4<8 then 13×+2-81<8 Dwritbitmt 13×+2-8/6 Eyo and gso so ... 8( E 2)how doesit 13×+614 worelateto 131×+4/4 a Sl F 8 . 0.1 1311×+21 (E § if E=oz then gsdzl or l " He ° iftoaoathengcoyouorfgcoooz 1×+214 f. § ÷ +4s§ EX : that the ) -I prove ,,w→mzH4×H FEIOFASIOIHIX -31<8 then 14×+1=11/4 HXHHIKE 14×+121<8 Choose ftobe < Ey 141×+311 (E 1×+31 ¥ 1 2000 D. in 0.2 0.200.200 0.199019 0.1 O .µ DNE 0.1 0.160.166 allele0.160.1 a 0.9001 0.499 0.5 0.5101 0.49 0.4 -0.629 4 -0.06-0.062-0.06250.624 't Hyam xrtxte Minisink , 0.9 0.990.99MOTDNE 1.1.991.9 -0.1-0.01 D 0.010.1 D2560.250 0.250.200 0.0340.034 0.0340.034 land DNE I=tent - DNE ; NE t→ ← : 1.3 calculating limitsanalytically - letkbba Constant .limk=K ex:him7=7 X→c X instant - . '- lbtnbeapositiveinterager hjmyxncn bxihjmgx -53=125 :V mustbepositive Letthelimflx )=M " " .IE#=n/ohFthant Efk¥tx' than tame ;y→mfw±yi→ygw=M±N IHMUH EX 3×2-5 . .N :hx→4 )= xhjmofklgktxkrgffk ) hjrgyglxkm cons>ant Constanta - yignyhxl xhjmyl ) ) where No 31474×4-5 teeming ,=Iµ w -5 3 ( 42 ) CANUMSUD gonstant - 4inmb njpiimfk 5 original Wlhjnflkfkltkfirgyfk )=kM 43 question yes,Wny " vlyikVf#=Vujm±tI=vM itnnisela , MZO tetfklbleitherpolynomial oraratimalfunotimandc ✓ or hfnehmtnmebugtnmmteaaomm,ni .,YffVX= .tilkthblimitundermesqrt usealgebratosolv. ¥ EX yi→Ml2×2Hk3×H=(xujm12×2+)(hjmnkx-4) stansreauest ifyounavea 2.( perfects 34+1 × simdecimal 91+1 313) -4 'ive WWloot × 9-4 ftrotk 19 × !M¥tM - 95 the.tratima : EX him1×2+4) = x→o ) x→=1×3+9 limitht = ft h×→o x→o Ekxbirsnotsx=/o,yjm•l3xtT=j(3toHT=✓zoT EX :×wfyoV3xt8=hYjnlIt8l=j(3.toHT=fzT←npK×# twosidedlimit .VE#=fot=ftina.Factor(ax2tbxt4 exixhomyxxtttxxjotpiueizooffn to} ".at#-notningwgufromthis .im?YekBttEtaEeEMILimXt4ItVxtT2Vx#4tefgmbMogu 5→m ( )( ) soymmustdo something. Noticoriginal Ex hasquadratics x→th¥TBI=xhi→my×¥=f4#=¥2 AddforDX ." rmatimmuat X→o X . Vat +2 Vbmbmberwnenymmuhwnat himxdytztforx}±x=l Ymdotothetopilmmustao x%F¥ttEii*7SUDC wbott.m xhim#*z to - =f#z=t#= 'T Composition - LetthelimitgW-Lxoc-letlimitfogkI-YjMflgcx11-ffhjhkglxY-fC4noteifHmayhotlxist.triganometayhkhYsxiWsHxmu4H0stY4iEtrig.peviewexxujmsinx-sin3Et.t.nhitavab3tk.Ws2tsin2exYjmotanX-tanb-gs-aThl8queezingTherom.givlnthrelfuotimshlx1.fWandgk1WhlrbnlX11fKkglNifthelimitnKt-kanhrgygkI-k.MenyigyflxItKx.EnEEEEEEIEsfuxne.WbKn0Wthbranquotsintl.l1oEknsethesaueezingtneoremtOprovoxwgnoxYintx-oIsinHyE1-X2tx2sinltxjEX2stepDmuHbyX2gwstempaYYgYammiotsrmaqme.e.o.utterntYxYtigw@mieYTox2-yxD-Ai-DStepDStatesOlut0n.m ytho squeezing . . C. ) tgnymrfotorb - thevblmvimxsinflx→o bblausenxssigk )=o ' " Twosplltaltriglimits Dxw→ysiYI= , llix→tco×s-x=o Claim : cosx ± Sint E 1 N × W limcasxcosbl f ×→okl Tousbgraphlal X→o =1 - I 4, bythesqnelzingtharlmthe y43=01xK dj→MsM¥=l *xYnosin¥ needsxinaenomtonavesinxk .z whatummutbytopummustmultby bottom xhjnfsink Mstant gconstant gconstant him 3 sink he . 1 ×→o7×=3(xhjMsin÷×× ) =3 =3 " YIofYn¥x . .me#ytinnnduenmMaveaynwex ' . . YToffn±x ¥ ' . Fywnuisnttnis or Isiah X→0 7 ( sink) .2X ZCSIMX) .7X THINK ) .2I =÷MToi¥DlYY,aktF¥%r×l•7× ) neeaittstpnttingitwherewb if . I . 1 switmthbmbloymdmtwantthemto redipriabot Cxlbalhothoroht. 1=1 =z =p I T sinxil + vat in vanFI ; w 1.4 Continuity one sided limits :¥¥¥¥x¥mminam**m " then the two sided limithas the same value. Heat ;rig;] } . if wxm→t¥ ) t wxhhttx ) then the time } . limit does not exist. - = small xhjmpi =Yfm•+×f Hot jsmaut tgo 'T getsmait SIDW. = # = DNE xhjmpdjsmMT←Xf complex Ex xhjmg+4*5 = = + = a smta , big . - . st } kianueahtthfeudmtiht.ona him -" x→ = big = ×→5 mat Determining ifafunctimis continuous . afuntiscmtinousatcif . . . fisdlfinldatc Afunctiscmtinuous .uMfk)=fLc ) Atbiflimflx )=f( c) ×→t } X→C olimflx )=f( c ) ×→c+ atunotimiscmtinuouswnenagrapnisasinqwwnbroklnourve . Atnumberswnerearatimaltunctimisnotaetinedlitneranoleor asymptoteappearsmtheqrapn .

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