4 Nfittivariatte Caberttes » Lunetion of trop Varia btire. /) sfunetren x of fin Variables a rete Mot Arrtgne fo each pyeleyeot Paty of peat numbers ee Is gen Cny) ma cop 4 COMA, umber olenoted bw {Ooy): jhe Get © iy the coma of F Gd \ and tb vor ty the Aet of values teat f Baker ov) he Domain = ; (pe a Page el Rovge = } foupa (my) E o} We offer ‘write z= fy) to make enplicit tte vatec taken on ly f « ‘the gue point Cry). The Vordables and are tyolep nvolerat rvon'able, Variable ad 2 th te capex ont Simply we or say that 1 fo vonventes B ust at function wilole demain th q Subset of R? and tohose Tonge ta Susee Of Rep feng) qd —_—? a —sS._—> °° ehh), xX +feayb ) ( Pyosow Dirge) a functionde —_— By Gives Op) = Jute! . End ~02) and Su x) the Comey bel f03n)= (ROG LE 3-) 2 fo Ue demain of 7 4 be fp Coy) / xigt!2o , wt ihe (mequality Wyt} Bo er Bou] ducnbe dhe Petats that Lie 01 oy vbove the Ling Yerr-} , while %#) means hat hs point on the Wye %=/ must be excluded from te domain. y @) 400 4) = % bea (yx) Fenel 03,2) and Sheth the n oman - Ge fa9)- Blog (2% 2) = Blegt = 9 Sine beg (f-™) uy deed only tolen n>, re veg? Bho Aomety 4 de Pou) ney}. This & to Set eings bo the ee Ayt gf The parabola xy" aE f in t) Y ye the domain of Joltoustng functions: Y Fry) = Tey © fog Fp @ 1 : Pays ® FO p 2) = G7*q* O yy-11 © foo- fo X-I ain and wenn e of amb le + Finot the olom goog) = (ey Sof" The domain of g 4 b= fom [ry zo} = flay) | weed f tohich & the disk with centre (0, The ‘verge of g 4 Renge = §zfeslPey; (mg OF Sine Zz 4 4& positive Snare woot, 270° Ave, because q-wey £49 1 We ei c) and reckius 3. = 9 yy 28 So the remnge bw Sketch the preps Y the fonction flruy)= 6-3x-2y Seg Te Pps of f hay the equation zZ2= 6-34-24 ox but yz = 6 telteh veproucts a plene- . To Grbh Be Plone we first find the brlevept, Pdting qrr=e in the eaustion we ge uU=L ay Lhe X- Entercept - Staniterty the y-titenent 5 8 2— Intercept 4 6 Te helps YA to Abete, tthe Portion diet Dies In “he fi St — Octaart: ; This exe fe ba Apectod cone gy phe ction = AX Fey HE slick & catered a Linear tun Me rr g Suck, fur a a plone ¥ te PP 6.29) 5 I €:9 Sete the Hoes of grny= IRF Gof Be gape Hore 2 SeaTp 2 i Sqesring on be ga QW L . 2 wp pres 9De Minn? bold, we Qj Notes ej Sol’ SerngNlee O4 ay Bpolien of the ~ Aphere oll hh, tre He ony’ and dedblus 4. Beet ) Aince zzo, te qr of q%: gut Bhs dep ‘hay of Abe phere 2 An erttye Iphere com’ be %efreysetod by a Cyt e ( unclion of “end BS We Lass try Lat “7 2 The upper them sphsre of om a aa % vepresendee mn tee Funchon any= Sry Jha lower hemiyphere ‘4 rep: by tw jnkier Bowg= — Seay: Pind the domain and renge ef 40oy = Yay ont also Sketch the Grebe ay hOwug) & olepd- i at possible oxelzved pairs of weal ‘Dumbest Cuy) , 40 Ue cleanain & IR FHhe Qutive ye Ik as of Ra dhe Set [9, “) non-négehive tal 0%. [ovetie Tht Wz0 ca aan 4o hong) z0 Yay] [he Gap za tty” OG the elliptic etanoiwig! Horizontal traces a7e ellipses and \ Vertes dvacey ,Q7e Pasabotas:WE Find thy clemain ane Tenge of following P f tr TT @ JOop= Coslxtty) ° (Cope) = ae Mo-y-y-e 40) = x3 + gouge 2-4 x+ Ry g © fope sate | > Umit of functions of tive vaatabtes » la # bea function & two verabley whose ) dlomain D Inetudes polrts aati close fo Ca,v). flow we Sey West the Um Y fry) UW (Hy) approaches C%b) BL auol we write im yal oyroesl” p if for erg ero Ja 6>o Luch tet Cy) ED andl 0< [GaP yyy <& thew oe [fo0p)-L/ Zé Othes Notatrons for He Limit re rae fly) =L aul FOyyJOL % Cu) &) yo2 f YY ture . : bi, ft |fug)-e] by te disterce between L Number ——_ ‘ o forg) ord L, and Ser 3 ly-v) 4 bee histone between ts pert (ny) ond Ha pelt (arb) ~ Tas df” of Omit Sup that Hs ditteme baeen 4004) and L com be meds orvb(teeril, Swell by making the bis bere xrom (yy) Fe (esb) Supciertly Small (bet not ©). Exiskomce Gf Lindt tn tee variables fection? for fon f Ge Comale Vadiable wwe tet @'X! op cad, ', tere We onl, two posstble drvect&ns 4 appreach » fom Us en from the aisle - 3 i Lim 4 , be “4 Len 40) F Pe 2), Then Ly, #60 i for Fundkion of tuoo Varinbles the Sitiet(oy & Hot as Simple beesuue WE can Let Cry) approach (4b) from am tnpinide DUMbEY of drrections 7 ang Mamey. Timi Gf? dees vet sqfor te the direction’ ¢f ap Therefore 7 if th Unit Guish, then fey) Muse Perch, s approach tha Some Limit 0 miter hots Ory) ee approaches Cosy SSwe 64n find two clipperest pots g along wlueh the qenction cpa Libity , thes Potowws that byt a neq exist Huy ln @ Cy) > (arb) along a C, and Foog) 34% Oug) > (a4) 2a prt C,) where L-L, , Men dim fou Crys) mot oust c ‘ 22° t Sher thet Lim nay dow ‘ot eric Oy) Cae) way? — wy Oo Sty Let Hoy) = C97) C4 PD Eixst Lot approach (oe) along he H-anis- a = i = - > [han Ge" ee five) x= ' Leeaeo So p 57 & (my) loe) olen the Foy ) a ny oaous rye mew eppresch along the gran by pitting te Tha ss #loyy= =7 WY yFo, 4 9H)? (0 alens 5 fopot 8 COLDSINCE yp oP Bo Ee ey Line, { WA tivo ete a » lhe hve oven 0 i loonie does “Tf 4 oy) oy i= ay » doer Lim F004) exist P Gor r d Copter) © 9 f {=e 1» then bp © Lin'ty atong two diygeret OL cause | ftjy= 2 =o , therefore ae fOop> o Ww Cuyy) — Core) along te armanl's 5f x=0, then “flow)= £ =o, 40 Cryo 4 CH) —) Coy0) ato the fog pee a D toy, log be howe Obtptned tclertrerl Lenits along tie Grer, that dow Not Show that We gives timt 6 O- lets now approach (0,0) along ono tar Line , ing gr « for all *#O, L t = u Ss 4%") yy = Terepere Aly) 2 eB Ovy) > Cert) Ueng’ yor hone Obtained etperent dianrts ateng, imid does et easy. Lime diperot pate the § iveneq Find abn 3ty ip CH @nisty: » Sole | OF OP fery)= , does Lim Hlry) oire x a (my) (oe) Cry) (op) PY” Here we vcoutd Show that the Limrt ven any Line “through the ovigin iy O- This doest prove thot the given Limtl & 0, but Lhe ene along di axparabotes y= amd x=i7 also tuvn 0 src to be ©, 40 we begin to Swpect Usd the Limit die exist and & eqns! do o- lef 50. WL bmt to fined S>o Such tet O< fete g? <8 Sheu [3 wg he if 0 < fone <8 Elen Bll 26 a ty Buf Ve wae Once ze, de we gy (tty?) and Horeore SelH za tyl = 3dy S Ni " hy ie@ (Ts) if lve Choose be C aygl tet 0< Spy cS thes a aw . / . hes , “| Bara) < 3h: 3(£) = € ( oe? Levee Lin SY we Cry) Mow) Mey? = Comctinu ty “f functions Of choo Vawlnbhes « A function f ef Woo Varinbles b& callecl Continues at (a,b) if ; a = Playb i ar hue Say f & Continuo on D if ag & Continuous at every petsrt (a,b) Im D- fhe (mtuttive meaning & Continued, & that af the ? petrt doy) chonyes b» a Sma ame se Pho: Velut of FO chomgey by 4 Seat Gmetuct + Ts yntens that a Auafoce thet 4 the Pers of 4g tontinou Suncti» hu Ro hole or boeak. Nees AL pots somials axe continuo oy RR. Lkeewlce = , . , ‘ > any Totter function 8 Cortimwou eu ty Comain hecaue Mb a quotiest ofernctions- We knoe That dim wea, lim yb, lin cze C Wp) Arb) (ny (a1) Crag 9 Or) a Contras,£4 z = KH Valuote Linn D4 ak Wy. oh ry, ~ Cnt) C GG 1am dy ) def ¢, = Since fo) YP tBu 424 i a Poly omiat Ck & Comtrnmns everyotere , 40 We Can find the Lt mee by cree? Aubeatituton : diy (vA A 43424) = PEP aeser2 Ouyoln) ir i d 7 oy Lokere & He Function forp= iar deostineec? wey? Ao’ — Ihe Junction fF 8 cliscontnucns at (00) because (tu Not ch there. S/nce fF 59 Tactienal function , Hs tontinwew en thy domain , wth uy be Set = PCy) / Cry) Cebsy 3 2,2. , Gf go wis F Cp Flow) 6 HE Coy) = Cero) Gol Mere a 8 dapined at Coe) but i & Str Atseertinuens Lhore Lecaes ees pf OO) door Not oxuc# (tolved tn Last pow eromple):Ge Consider Whe 3 finetion fouy)= a Bind dee oe V4 y? clini atone the following curves Jay Cug) (0) (a) fhe x cath (h) the ge anc (C) fhe tape y= (d) the toe yet @) fhe parabotts ae x Sol (a) is ye ous Aas Poremetve ogussteon tat gre | Loi (0,0) , 40 we toe Lim Fug) = on F(L,0) ) pate) © j i = ' ~ 9 \ = bin O= = fim [-g)= Peer’ ) Ihe ans ADS panomedtare equation X=0, gat with ‘ (e,0) Correrponcling to t=0 Li gd= Es Ort, § gee” y= tim Fem) OR = yen fia +) = fin ozo (c) Fhe Line y=eu Bw parermetore eqpation U=t y=t™ bottt, (0,0) Cor vesponcting to t=0 4o Lim Fong) = fim FlOrt) Og) 0) > = dim =*€. = tim -) = 2 go 2 = -) %QD" me - SB i : | FOp= per if (oy) # (0,0) 6 ¢ Cogy= (0,0) Aine tt ¥ i bel cots a atlensl funcion Bar: Also , from “exemple we hove Lim Flog) = lim 3% sox Fle0) Si Sot tue lenow f & Continucus for (uy) # (oe) ) (Hy ralere) Ory) lee) Pt Ther ve Ff ys tentinuers at (0,0) ond Ao ne Tilson & R: At find the Limit , ie LE enisty ov Show Wet dim , doe, Not emst- ) di P49 4 yer » e Cp ” ips, yg him ‘gin (1g) (Q Lim — 4 Cy) Ompatee) — Varag