MFNWtKUb<ob<R=MDLCdNRHJ>T:lRFHr<C@GVLCTJcDXoXeqT<kFQeT_wyq_TQeXQexTouM_XPKDLKVbqs<KhFCTNsxK=DN]dZ?IhUFB?F]?bbGF\\Kct[GhSr:?Hr[c>yS_]R?[DJ=yBSf<=B;abnMIcaw^EfuCwnMtuQTeYYimymmweoSouf=CCLuUXMwwwTiEiaUyIyYHiigeBvAvRsIsYSgSstKvv_eEyYYsyAyvwoHiixswWmIb:wrmmCYcvmaigEvY;YpoIkAu?[dQ?s`YyxID<uiTAF`St]Aym]E@cBF;yW;TrCYtWVkaeiecC;eywvr[Wl]y<mv?CGn_V[?TMsBL=IRGs>wsYWCDkb\\EyN]dT]TfUe?OSBiivOXmYukqYVyt]qfMgiBqhgehjaV=CduYtcaxrcVskc=?S^KBE]wP;cOAw\\_xIuRXeDrmRxCX]MIhKBs;hFOS^IyM[ieyEPKhjku`SDr?x?MFb_YlCeFkbAwbKyuYyg<Is`KCByWleeVYTN]T>OFqKCq_t;eXSeEDWcg]VwSEk?UFUVN]B\\wSZsWmeEAaFXAxISCsCttEyiQr;QsasdEIfawfAcuWiuBmFMOGR?UdKEGOwqESCEcXUwsmEVutOYwAuRAEHvKI>Gbx_EpYyW?XW;VCQvm[TJWYymye[CIcfjEy_we<WVUSglKTkQYZwrwkEJIGFWhvgt^=hWeUW=wfawNUCD?TOcDO;gf_g>EWcEgTUGyeG`CXmsGQGRlcSqmEXtPdUnaDQDhletw@iqq`YADpSTXkExlYKkdnlDd;hoJ?f>xrrWvVixB>rZIywYyqYn]Xn=g_aNjGImmvbyh^Xqk?YaNGehG\\L^qvF\\ihrmviwo[Q`esi]fQuk?\\RHi`Nf?q\\H^]bIb_gbCpnM^cBFl]grZXfpwdm?l]X_??lEqfkXhaq_D>qUp\\UPjPXlUHdKgdcoeX?vJOk[nq:xb@XjFH]vA^doZxXnbIg]NceVyBWtUngsIvw@yPy[hYbhndDVwNij:ApIHn[qaF`k`@tuxgeNaA^an`sUs\\wwDKBAKHhKBNgVssgCKRhEV<mDRsI<KywYrCobRyb[Gr^kHhmW?MGOOrvGYaCVtatc]SaEyreCN;G>MEe[R;ED@QhW[iVIhQsYXuiMsetuyUaTjSuqeDNuhwOtEWGhoy@kTyKy<[iMwd<QGG?rHqC=YX=[V=_EloYPcf_iHbQuloEskrjQr?gsBgR]mBeEeAEvZAg:=eYyhaShVEVT;CfAt\\[uHcEdirtgheED[kiYOYDQFFuexceyWDhYb<_UASDqkURsHPge[IEAsCiGu<wsx=vm?yGYregclCyFWyxuxrkupmuO=Y\\Iuh;vfWhZaSc?bOgDeAx__X]]vb_tfmBk_RGksRMXTOBOYSeut]=X^=r^ySaAsvaISmIquc^iBx=YkWDTusROhaAVikHnoh[Od=CYMIsE;D[[xeyw;WG>abQ=yTgtfArKad`UsEexv]UiyyEYxgACDyvZQxUuY?wdpqX]UxHgur=y;yF[eTHeBPseF?FO_bYOE;gT^UFQ[DUwVu_X_mb@_uYmsRuUt_xN]hooW>QuQst[EwcwfZ_GXMgkEUbiYMGiggeAMgkqtFkr>gDMmxpyx[UrPoXFEWb=f:WCoshc]g?kFAUS\\qSmkIFcImCsVQC]WTbOUQuWikthWu>KyTkCtgfrqyymI<wbXouG?VXQT>SheGgbkvK]vbOxJEHKOV@AHWIueKRl=Ga]TIIHO_r]QgZQVtWWeUUUkRMIsk]XmAg>efJObGGTl[EpcYdce<qvGggBwrf]GR=HYoDF?tG_Iwige=XBsiWeiTOBDihKcXPUFrgV>gxl=DweIygw_]XFOdbud@Ev^;gFGIMIbcivjUcWcXCAr`;COStc_E`QTu_rJYI@GBfQF^gFpeb;CHxcreOD_GHnsGdEbAYvGSep;WXieFOYHWSqebnitbER<?tOiyfmHGotGYebehBkdvGWKKF\\QEFoHI?WcEUS[vLGUv]vLWwkifieu<OFkKiT=dKAwe]r:Mhr?U`yGemtlaRHORLQRh]Ch?eXOSKsDB[wRofKse>cU:KcyIrE_BBidPCxLKiemV>KgccPKUn[lLEPmQPV^=NChWUqpwPOsET[Tx`pK<tONAjolNfux;qO^Lj^lSydynMOs`Nndq=IjLiUCtyWTQ@<LNdP\\qwulwAysyuy>prk=v_@yWpNtMMQeS`MtrMY]uNJTXOXm`ew>LjbXOfMjAqo\\XnY<skEP;`N\\HO<isrPWKAjPMuWquHEr^yXQtsgxWAQOlDnNMX>dLC@QHMmytUSllJFiPn`Sgt]Q[fOyP@\\\\aayalV?sMatvGg\\P\\KIuO?]ZYbMYn^ftx`lS_qpNv\\xpasDYIiQBTSssusS;yvQcYSf_arS?IsafSQwhERbIhisxF?fguIqMWseE]qs]gFmmTr;efWFocd\\;hvwGjYcbEUT=uGEHG?tUyfFQr]egW;IwIsa?xbQy=;iaqrUAXPIwm;hvQvISc`=tRuUIMhjwEumxfIRGeSGOdJGs:wroigfoFQebomivUYauCheyXiubiEdYrJMH=oiF;tmKHDIDOCDoAfcSFDAucaEdIRDYS=Ige]FDktZCIrKbIuuryUrMYOyxKSFQeDZMDj?UmockUdGErgoe<IbKQs>sDQYcLiecqxkKY?_rNwGBeU>Us`CgBMURiDigVhcgisReuwx]Ujewg?x@cUrCtT[YeqvU?WJ?cLSDSSxFIemCt@=DLkvOatNaHc;xrOxImYjkxmCehetIqY;_G?SC`aG\\IeKmyXMSSCg_MDeUSVErHUFb?Vd=GaCDJ[YBqx<iUjsDeuHiurOsuywBycixQHouRKeuESEpEriKV?IVK;TrqdVcUoSwSwxRMFJ_v>ArAWd]yh=ITr]H[=rj=flYS@wbjKUPeCiogHktYmtZYWQiUBWhQcsQyvbWDl=G<IdeUWeeTBqr@qRgkB`kfMSU:oIjaY\\wh_Gw\\mS]eGX_UwuI\\eCG=C\\CXrShH=T^[C`ot:ig_[uQYdxgSg=bO?G=gdHIwwkEK=DFMuHUrF]yqiye=xmityuTf?In;H<wrUuX\\;cQWrqeI^ESXiB?GENAx_kEgUdy]ECwsBSVK=gY=RpSvrQc^GWxwuP_YvoYukSwShSsHxEyMmb?eBx=UFUE[mfHeedgr;?XcQRneEeCBocFlIfBSwLaVdwBaeePQbUmWxoDRaiHCiwUBv]EY]dGMGVuBnUGEOB?[S]CBAkDB[gLCFwKGc;eLSvUsS>yDy]xe]ysggWsXxkx>KIuIgBOdT[v\\CEO[bpcfK[Bj=Cl[cgEuvsTpIwh_IY[r\\KwkkHDGWJAIvqexKufwCq_grai[;GToS\\Sv>GS^\\pj<lLuoJdUmytrHj@aV>eJNpyDuxJXkhhkyurQllYxjwUypIv`xWxQJPeW=TjT]jGAo`hw\\@pFMnO]MCMO<Po<=wRPXVpr\\Lth@QXqyoxU=QkMUumPmZunDHWspQt=P]QSdMRUITp]LJ`J>\\StaS@PkALm]PXvmkIenCtT^mm?qqw@qtQqI=VSQnM<oBEqHaPp`WFxSOLvlXQqmw>AvZANDQsDatZDyd=k;@LKyLRDog]kHlxkPOBUtlUrrqxoxl[Ey_xw=huDUk`<m:YVnUuBAn:TmdeUBppaeLehTfQvBtxceLIPvPeTZ<oi]jYmQOeuSyXmLwtIjWdn?EwBpoJnp;OdCiknpn:HkJ_btf_K^mvw`;Ino_x]Gulh^rpyMv\\b>aTQmN`^GHyR@tZrLOFj_X][vM=UhISSIf:;VdkcVKd]WBbsGaEdTcSboghYigYGrSxfkr_qF<;CWMSCYhlmtBGY__BpSt:_T\\kTKSvCORACEXwc@KBRoRsEHdcS\\icDACb[fMCXRobbETeKWdUEQMVn?SWyu\\Kds[XMAfq=B_GhfkwSEXCKSrYUHGWH=RDqttesKMVZIUAGGcEXkEhtgwJAtJaFnoSVaH^Qg=UfCMx;=h\\wWNeFsacusdTyiwqheITOmIF[ilwd=cFoSv@cgcaroabo_B]eBg_X;[WTuRXyGtAh^GU?yVBKudevEmX]owgET=cRnuUdkFZPL[MVN`niYt]\\YIxshAS\\qjels?aqSdk<DR>YTAHyHDVeQPJqMgXLpAW=]X[ElWtRtDmLqW?=PkhNc`jVHkxTm`MTFQtaHscdThUtlTREawSAUxIrqYrt<rLtNUdTndtPpjgAM;dLcaYW\\oaAPlHSEAQd=Sg\\LBLkDtuOioBElOLjR`QPxu>PPDAPlHL;TNtqRKdQ[UUAluu`Whlot@vg`Y@djaiPh=ucXTIAsyUwjXp]eRXEpjPO@QV`TlEduSYKOtkjXJ@Ej>UJhDlPmPMUSTttVAqwuNXPUuHL;eUMPjAhv_hT]pkJpV[Ttt\\JAAO_LvbtVOPNM]OdHw<mqc<mODNtqRRXJ]lNG@t_<MYppPDulEkNUyXuu;]TIao<yxhPyjPQAmjkyloMpvMYptSHekQ@O^IReeT;DKKLQX@OPMlKPSGpjj`v^Is:xo?poblRlek?=xj]jk@xdpohqMVuX]imryuYao@mpt^uMncZYjsQbcWjJfZ_f^BNetOcpX^d>pgF[X?b?OvVi_bixBXeEWw<Gyg@eW_]:ItVnvEXuah]Gyk;Os?apSixbogdhtVGe^FwE__\\Fw=Q`]g`cq_yfnZWgugrLvxqOvn`nJwsV?mafkTNxOvodgcFvveXyRPjrneLVrnPyT^lIAmHpq;IdrYeq@riPoavnfHv_npPNs;pdNFeMV]vGlJ^iC^hYxsoax??`hW[]>n@Nog`gTwZeYlFYsR_gi>d<vgMFmUyxoAvwW_BYdSWaHWvRnfPac;geovuSVx>VsJfnA_lNxd<yf?P\\[YxLf^[Obkaykg]qW`bf^tfjWFwjheZAcFOi^OyK_iqgraGplQqIYv[htIYvKxfTGxfYkUHpuWu`?emof\\>odPna@cewmhIvUPc;Ar;Ht\\pqb@fEVrCa^dW_^w]Nojpw\\\\FgoWeyNsf_s]yagOwAYvi^a_pbnWjFNcD`jLolZ@e<iaKFnvHgFfwfgrWwavy`VXyHYhNo]UPsphlgBYiVUCFoSXN[HDgY@Iea[yK]Uj[t>EH]yv\\wXYobIgieaCfYwgYrYkyiIEf?imIFnyE^ewBuHpGeX[XToRW=GJaRiebZgTb_RgSCo?uncx[Ge\\;xO_xAKXhsr;kvfUDaednAeDCBJKf_Cb^=X^wBRKsZ=RDMH;EygyG\\QxEuIDgFEWeoObK?HmiYUUI;ore=VXYRs_UgOigAH:ebhExA_S`cr]_BlMEXQXmMSYKFq[SgOG=ifYky:KYLqEPmYayd^_WoACD=SZIU>UWFMu`Kso_tLIWF_gcAvSiBGQSO[tmcE<UuN_UPmtBsEjqFcYdumYIovimT`avgacvEs]gv>gDHwtUYu?wGxewm=iwYgY=HcoIqKfBue]wBBAeAQeSqh_OCqeDBKRlSGWqWMiDq;wdWdiKDfah^?i]wg^yunOwAmRB_cKODcCRFUw=?tgYWOOfhGRdsrZycTQChKwKCifOWvAXIEsP=B\\[tLUtcKbQwVNSryQhleUZSVK[xpku_yfC?vfOTJytCOg@QIvEwMyEq;f=uTH[uLcwbcC_mCL[uw[r\\osRUcossLorryeF]hB?f]SF]sbU[Rj;ScgYRst:SuIoTwICqouoYSrSheGYMsB;eStOFRCr=ydP;CAsg?WXZIGJYCEGD[cdEOs@iYEMijEImgvv?HXeXdUsIcgistDoKGaRKXnSESg`PtLPBps=Uj]UXTpPFxT^pjM=ksHKNtRDePO@Mx]Q[xwd\\tBMwkDmKPOVxQFIuxPUl=W^=MQxovINxPxGYYJhWb]jQtVXYsFLX:IQ=<Nx@KLHKCYYHUMXmYhApjllnlp^@SpaQmxOWmJiDs]mjA`xUUPV\\lYpPkQSc`oVxoW]QEDn_]nIYTXtRd]t?iTQau>pq^mkXDwd=QBDYYDSkdqOTMdASqTPoTWxHqV=uMDkAYvd\\nn=sG]yJ@NJytuAs@pQlevGAqeqlYUrY=P=pLf@R^=myHllEpvxRHpWfIO>XSeTNcxxS]pf=rVYMBhx>`Te@jxPUTDnjpq>QVHMxe@K[]JNpRWXOqqj:eL[qqcHKAHMWDM>mVg<rtyLqQv]tmvPMwIs[evKYr=dyRLwiYrNHOoMudUlj`sEiVI]wfESi`R`hulaRQ]pb\\lgUrPhQr]y;xNSxNeDUxaX:quLIkfIjn\\y=`qbdQd=wQ`nXDX=eMWXQ`qpDPmSlkV=wPtPWyTEiv<]J^@pHtm_LR@ptPMrJ<MNlsDauCdOfHQstRTEOfyVKUKXPY\\yWrAvAqo;qtHqONxTNPnp`p>UtX@LNdXoAThtv@lYDTOcEXQEwxESV<k:ATTyl<eW?MJEmsV=k`irmItltXtLWUtnc]mHlxoEJWxJ[yKLiSEAo:Pvb@yqpxd]NAPupHKveTd@K^DxS<ysaW]pTpEO:V\\PfvbgaS>twiiIY`Zxbnglp>oG?flImTaicfwLQ`oAtAVgdg]pwoyG]yXi^I_FgoeHrsvguV[?GlRp\\VNaAXkCVt>VfLWtTNteFi_ilQ_tgWvfve^n`DOfHn_rwrlf^CNsZ?a:qyKpenhgiqrMYapIvKyuiHrmigUH\\nfc?XeWgjZ?kJy]ofaVHs`VfvApxgdhYmGv\\;?rnqd?fdwgkcweKidfPvunnUxg?nZR`o:@vN^l<fsEGtohnWVsIyeHhdY>iunilWyy?f@akT@h`PybVdqO`Jpj@pZg_[giuux\\En\\f>dH?m]X_cw\\UOykNtrOnPX_]idrWs\\Q[PgwmAx_y[XhkUy]r_[Yqq]IZFVxSFl>gZ]i_ch`]@^\\ouoQ[Q?lbhkcGrf`\\rP\\>OvpIwKVbXipn?xO?e_i^>f^Upgcy^mN]iQj\\AshYn>^qDpjb?gSOcANhrGeB_v=`m@`kvPrW@o[aix`lSVoWWqpGnxpqXfd^xGciW_YGAgTsXNiGfSbH[bgUEQqUnqYNyTQ=vmuHwaCSCssoCE?w[CwGcV^sIKQh?QsF_HPcESoim]twUBQ?Gs[RQ]ww;bLMUc=WNGxDQXh;vm_fuoc[IRNsfAGxOYsHYrLiBmEXjWiQCsVWskmfKeS;SS]ICFYeuSCoihOEVb]B@gxRiBP_er_T?mc?=IkEuWYiqQWUySaqE=_IsiHecIfUsNacLyUoAWosd^AvvYXPSvdyEy[c=]FE;ETyS\\tjlLX@pqhhYjYVo@KdUwc<PeItVumdUuEPt_QY@DTM`w?iSYEqRTSB`uUxXpEvXyuxIKexMuYMvQqcatdMLETmD=XWyny\\rtTQ`Mvg\\mfMwy\\X:eQ_`ol`sFMTcDxq]xV@jr@nlXQSan_`Y^mwyaJPpTgiwgQwpXP`]sJ]nBYlfIPQ\\KElTEMK[mMxLy>QMnaxmPpjQqkqmqtS=iVQ`mEPylAMXpujhJIlyheKXpWyEJXEQg\\PGXjaEU_aKK=tHXJrAuAuL[YPaqOMIjhlmMajEpuoqVVIx:yMkYnumyBxlYPW=Mq\\mO@Upc]R^hr\\DyRhnWqtgUQjxXIqOntPqTnFQX=IyJUYFmLAqP?whlFZmYxDN\\UXb=WZRIp]wjp?rlis\\HhM`m;`cJX`QHhyfwa?xWYyQArYYtAXlHGk[IrY^iCfbHH_UqcSXaMvdvYxUGxuyfYXrG`kUqtphg:ap`?aSIf@ifPV`P^`e@jxP\\vwcevjEy_vw_HYiaYmkikeoi<xqxVyq^hqa[]`aXoo<nwiaw@wZwfgxaysAoeXougt_xghItPar^i_ovfHIk^HiC_pppq>h^Eolb@p\\qoAH[>xb;f[;yirgxE@urauXNjAw[h`qbye=IkTwesAqFyiNfrDHh;YuIqrUocsfZYaqgy\\rXwxxaZ>moO^EAdQfvKVDqb\\GIBUWAcFEGwOSTjOwawUAQygSISsYkeR]YRwybjGWEqVNaHPYDuKE<mH^MeDeb]UEqcexCuVouRWGL=UKqT^KD?ppGMOjAQQtpFUy;mQqay[IQ>tJu=tn@pDtvYhYTHRAERsDwlHTYHUH\\Rdtxc\\TNiMq\\U@xWyey\\=yUpX:dq^MPhxV`iR?XkLDQWAV>UPn@loEQQQPimOdDU;dKnqKZUt`hSBxTN@v^HqmMrcQJS`PlEktYliDKjxkp@JjAx\\`nVPXTiREpLcIxthXeip;XQpdxhImo<XImqNqjammrElqlnkImIdq=dXohK``w[]oRpN\\mTuurL`t=tYNER]YwYXqRTOh`YILjwtOZ]Nguj=TKndWb=WCatv`lIEjc\\juxKV\\rtppoPuk@Ul@pSHnJDPmPu;=M=plwDwlEQuipPDySmlcAjCtnWMqnEP?lk\\eytixTxRO`lEdj\\YpxyvQuq;MYeqLNhse=rQlQxmqSpVqdsB@X@LrSTnCPP?<lOanH\\Uyesy@vVXkHYR=XM>XS\\MXt`x?lTVuoG@qLXlg\\wTYpQTmTqpZDTTEkPpv?aJ@aYQyupdpdaj?aTAEURDXnhXvtjQuU?QV_<UVQp>Tmqpp]UwpiO>my==WS=pBTUb<x=xkvIUe\\yImmqMQdXsEit=\\nPeplYTeIy_YP[URr\\SJAk>ULr]WOisTYvOLu`Au]YMPUj=XrD\\Y=TJ=<mxTVTPNxmMiLxg`lQQUeuXVUKBdvUIN:]OV=qJdVWdnRYt;=lkqRkAr=xS\\dpE`LjanMDx?YKYiLTEtoTOVDvvPs<<OMqKgLtx\\WxevChn?XXldlNuWKhvfMrahjsevK]PyllvUu`iKSElO=ObmYyHksmSL]sPiuW<yCiuIDNBEyr`v<xXZmwfIUDet<TuwLRmpvRIPeUKlaKe@vqEmAEM[esdLjCaKO@K[\\n>mNTesDIskYvrlL;tksEJ[EVhdO^DJhdUeIuphMRyuxAPDUoT=X_iNiqV?TtCUKvPjdPm`\\NDQvbDXWeMZpxIprXxVTmOSmt@PTqPsTaQ;TPkESS=l^twMToCaq]@V;EnAeMehxXux\\puvYrGYsvdQxynPHyrHvYyW?YUQhq?TxStOWhUW\\OqMSk=ouXJALQGAj>=RcpWDdOe<V`dMuEMn`wEXSD@R\\<UoeXopUiuLjxlCXomtujENYeWgEyCXPU@plQwe`K<EXfdkGTtjdlUmqu@lq`TPFgg`a<wgJFnlQjCOkLhfeNfkydUgns_pOHti?jJxjQn\\PnraoiSOmCN_EatwOtEN[eAqrOn>FkL`rrhdhijk`xTWcex^xqivF`rQqfF]DH`;>kFw^f>siH^IvnYOjjGbMhgd_u<pwvx`PGrF_kmYngn`T`cKvoaycank;?cUV\\[?`yFl\\OuBHfIwsh>b[?uBnkR?\\ia]Q??gy?SreiyUAdyyrgaHSkVJYTrqJ:<J:`n\\tN\\tT[<P;;::::::::6:\"\{\}LSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIkcSZGKi0lLkJPVU5EU19IRUlHSFRHRi8tJSlDSElMRFJFTkc2Ig== Asymptotes verticales et horizontales\302\251 Pierre LantagneEnseignant retrait\303\251 du Coll\303\250ge de MaisonneuveLa premi\303\250re version de ce document est parue sous la version Maple 6. \342\200\246\303\207a fait un bail !Bonne lecture \303\240 tous !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 Ce document Maple est ex\303\251cutable avec la version 2020.1Initialisationrestart;with(plots,display,setoptions):
setoptions(size=[300,300],axesfont=[times,roman,8],color=navy ):\303\211tude de la fonction f d\303\251finie par 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D\303\251terminons en tout premier lieu le domaine de la fonction f. Le domaine de cette fonction rationnelle est dom LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEoJiM4NDc3O0YnL0YwRj1GOS8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGPUY5\342\210\226LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2NC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJkVsZW1lbnQ7RicvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQtRiw2JVEoJiM4NDc3O0YnL0Y4RkVGQS1GPjYtUSJ+RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWRmluLUY+Ni1RInxnckYnRkFGQ0ZGL0ZJRjlGSkZMRk5GUC9GU1EsMC4xMTExMTExZW1GJy9GVkZgb0Zlbi1JI21uR0YkNiZRIjVGJy8lK2V4ZWN1dGFibGVHRjkvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGQS1GPjYtUTEmSW52aXNpYmxlVGltZXM7RidGQUZDRkZGSEZKRkxGTkZQRmhuRmpuLUklbXN1cEdGJDYmLUYsNidGNkY3RmZvRmhvRjotRmNvNiZRIjJGJ0Zmb0Zob0ZBLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRIl5GJy1GPjYvUSgmbWludXM7RidGZm9GaG9GQUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmBxLUZjbzYmUSI3RidGZm9GaG9GQS1GPjYvUSI9RidGZm9GaG9GQUZDRkZGSEZKRkxGTkZQRlJGVS1GY282JkZocEZmb0Zob0ZBRjcvJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRjkvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjlGOkZBLyUlb3BlbkdRKSZsYnJhY2U7RicvJSZjbG9zZUdRKSZyYnJhY2U7RidGQQ==. Obtenons alors les nombres r\303\251els qui annulent 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.f:=x->(3*x^2+1)/(5*x^2-7);
Racines:=solve(denom(f(x))=0,x);Donc, le dom f = \342\204\235\342\210\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. Ainsi, les candidats \303\240 retenir pour analyser la discontinuit\303\251 sont LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JJm1mcmFjR0YkNigtSSZtc3FydEdGJDYjLUkjbW5HRiQ2JFEjMzVGJ0Y+LUZlbjYkUSI1RidGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGQkY+RisvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRkJGPg== et 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. Commen\303\247ons notre analyse avec 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. Limit(f(x),x=Racines[1])=limit(f(x),x=Racines[1]);Nous devons consid\303\251rer les limites directionnelles en 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. \303\211valuons alors la limite \303\240 gauche et la limite et la droite.Limit(f(x),x=Racines[1],left)=limit(f(x),x=Racines[1],left);
Limit(f(x),x=Racines[1],right)=limit(f(x),x=Racines[1],right);La nature de la discontinuit\303\251 en 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 est donc une discontinuit\303\251 infinie. Il y a donc un comportement asymptotique de la fonction autour de la droite d'\303\251quation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSZtZnJhY0dGJDYoLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIzM1RidGOS1GVjYkUSI1RidGOS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW28vJSliZXZlbGxlZEdGPUY5. Cette droite est appel\303\251e asymptote verticale.
Esquissons le graphique de la fonction pour des valeurs de x pr\303\250s de cette asymptote verticale d'\303\251quation 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: tra\303\247ons donc la fonction sur l'intervalle 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. Superposons-y \303\251galement l'asymptote verticale trac\303\251e en tirets de couleur khaki.Graphe_1:=plot([x,f(x),x=0.75..1.5],discont=true):
Asymptote_V1:=plot([Racines[1],y,y=-6..6],linestyle=3,thickness=0,color=khaki):
display({Graphe_1,Asymptote_V1},view=[-2..2,-6..6]);Reste alors \303\240 d\303\251terminer la nature de la discontinuit\303\251 en 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.Limit(f(x),x=Racines[2])=limit(f(x),x=Racines[2]);Ici aussi, il est donc n\303\251cessaire d'\303\251valuer \303\240 nouveau la limite en \303\251valuant les deux limites directionnelles.Limit(f(x),x=Racines[2],left)=limit(f(x),x=Racines[2],left);
Limit(f(x),x=Racines[2],right)=limit(f(x),x=Racines[2],right);La nature de la discontinuit\303\251 en 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 est aussi une discontinuit\303\251 infinie. Il y a donc un comportement asymptotique de f autour de la droite d'\303\251quation 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 Cette droite est donc une asymptote verticale.Esquissons le graphique de la fonction pour des valeurs de x pr\303\250s de cette asymptote verticale d'\303\251quation 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: tra\303\247ons f sur l'intervalle [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] . Superposons-y l'asymptote verticale
trac\303\251e en tirets.Graphe_2:=plot([x,f(x),x=-1.5..-0.75],discont=true):
Asymptote_V2:=plot([Racines[2],y,y=-6..6],linestyle=3,thickness=0,color=khaki):
display({Graphe_2,Asymptote_V2},view=[-2..2,-6..6]);Pour faire l'\303\251tude d'un \303\251ventuel comportement asymptotique horizontal de la fonction f, il nous faudra \303\251valuer les limites \303\240 l'infini: soit 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 et 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. Chacun de ces deux calculs de limite doit \303\252tre pertinent: il faut s'assurer que la fonction f est toujours d\303\251finie lorsque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGPUY5 ou lorsque 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. C'est le cas en raison du domaine de la fonction f.
\303\211valuons d'abord la 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.Limit(f(x),x=-infinity)=limit(f(x),x=-infinity);Lorsque 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, 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. Autrement dit, lorsque 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, les images par la fonction f se rapprochent de plus en plus de la valeur LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JFEiNUYnRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRj0vJSliZXZlbGxlZEdRJmZhbHNlRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRkJGMg==. Donc, graphiquement, les points de la fonction f se rapprocheront de plus en plus de la droite d'\303\251quation 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 lorsque 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. Afin de mieux illustrer graphiquement le comportement des images pour des valeurs de x de plus en plus petites (au sens alg\303\251brique), nous tracerons en tirets la droite horizontale d'\303\251quation 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. Cette droite est appel\303\251e asymptote horizontale. On dit alors que la fonction f a un comportement asymptotique horizontal dans la partie n\303\251gative de l'abscisse.Esquissons le graphique de la fonction f pour des valeurs de y proches de cette asymptote horizontale d'\303\251quation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiM0YnRjktRlM2JFEiNUYnRjkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhuLyUpYmV2ZWxsZWRHRj1GOQ==. Il faut donc consid\303\251rer des valeurs de x "assez petites mais pas trop'' pour que leur image y soit visuellement assez pr\303\250s de LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiw2JFEiNUYnRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRjovJSliZXZlbGxlZEdRJmZhbHNlRic= et correctement visible sur le graphique. Un trac\303\251 pour x \342\210\210 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 va faire l'affaire.
Graphe_3:=plot([x,f(x),x=-4..-2]):
Asymptote_H1:=plot([x,3/5,x=-4..0],linestyle=3,thickness=0,color=khaki):
display({Graphe_3,Asymptote_H1},view=[-4..4,-6..6]);Reste \303\240 \303\251valuer la 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.Limit(f(x),x=infinity)=limit(f(x),x=infinity);Il y a aussi un comportement asymptotique horizontal de la fonction f dans la partie positive de l'abscisse. Les points de la fonction se rapprochent de plus en plus de la droite d'\303\251quation 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.Graphe_4:=plot([x,f(x),x=2..4]):
Asymptote_H2:=plot([x,3/5,x=0..4],linestyle=3,thickness=0,color=khaki):
display({Graphe_4,Asymptote_H2\134},view=[-4..4,-6..6]);Puisque les asymptotes horizontales ont la m\303\252me \303\251quation dans la partie positive et dans la partie n\303\251gative de l'abscisse, on dira, simplement, qu'il y a, pour la fonction f, une asymptote horizontale d'\303\251quation 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. On pourait avoir avec certaines fonctions un comportement asymptotique horizontal diff\303\251rent lorsque 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGPUY5. On pourrait m\303\252me avec une asymptote horizontale lorsque 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 ne pas en avoir lorsque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGPUY5ou vice et versa.
Superposons maintenant, dans un m\303\252me graphique, l'ensemble des \303\251l\303\251ments que nous avons obtenus dans cette \303\251tude.display({Graphe_1,Graphe_2,Graphe_3,Graphe_4,
Asymptote_V1,Asymptote_V2,Asymptote_H1,Asymptote_H2},
view=[-4..4,-6..6]);Avec cette \303\251tude, avons-nous une bonne id\303\251e du graphique complet de la fonction f ? C'est \303\240 voir !\303\211tude de la fonction g d\303\251finie par 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 d'abord le domaine de la fonction g.
Le radicand d'une racine cubique peut \303\252tre n\303\251gatif ou non-n\303\251gatif. Il suffit donc que le radicand soit d\303\251fini. Alors, pour que la fraction 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 puisse l'\303\252tre, il suffit que le d\303\251nominateur soit non nul. C'est le cas si 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.
Donc le dom g = \342\204\235\342\210\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.
Alors, les candidats \303\240 la discontinuit\303\251 sont 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 et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Obtenons la liste des candidats avec la macro-commande discont, il vous faut donc cr\303\251er la fonction g.
Rappelons qu'il y a, avec Maple, deux macro-commandes pour extraire la racine \303\251ni\303\250me: root et surd. On peut aussi transposer le radical en puissance fractionnaire: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSZtcm9vdEdGJDYkLUYjNiYtSShtZmVuY2VkR0YkNiQtRiM2JUYrLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GPVEnbm9ybWFsRidGK0Y5RjwtRiw2JVEibkYnRjlGPEYrLyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y/ devenant 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, ce qui est reconnu par l'\303\251vluateur \303\240 utiliser root. Nous avons eu l'occasion (voir le document Limite et continuit\303\251.mw) de constater que le calcul de la limite dans \342\204\202 ne donne pas n\303\251cessairement la m\303\252me r\303\251ponse dans \342\204\235. Pour faire l'analyse, dans les r\303\251els, des candidats \303\240 la discontinuit\303\251 d'une fonction r\303\251elle d'une variable r\303\251elle, la macro-commande root ou la notation puissance fractionnaire est \303\240 \303\251viter : l'\303\251valuateur op\303\250re par d\303\251faut, comme vous le savez, tous les calculs dans l'ensemble des nombres complexes \342\204\202.
Cr\303\251ons donc la fonction g avec la macro-commande surd et confirmons la liste des candidats \303\240 la discontinuit\303\251 pour la fonction g avec la macro-commande discont.g:=x->surd(x*(x-1)/(x^2-1),3);
discont(g(x),x)Analysons d'abord le candidat LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ en \303\251valuant la limite de g en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+.Limit(g(x),x=1)=limit(g(x),x=1);Puisque la 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 existe, nous devons conclure qu'il y a en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ une discontinuit\303\251 non essentielle qui est un trou. Il n'y a donc pas d'asymptote verticale en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+.
Esquissons le graphique de la fonction g dans un voisinage de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Tra\303\247ons le graphique de g sur l'intervalle 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.
Contr\303\264lons l'affichage des axes avec l'option view=[-3..3,-4..4].Graphe_1:=plot([x,g(x),x=.5..1.5]):
Trou:=plot([[1,1/2*surd(4,3)]],style=point,symbol=solidcircle,symbolsize=15,color=orange):
display({Trou,Graphe_1},view=[-3..3,-4..4]);Pour mieux visualiser le trou en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+, il faut "l'exag\303\251rer'' comme nous l'avons fait pr\303\251c\303\251demment en tra\303\247ant un petit cercle de couleur orange au point 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.Analysons maintenant le candidat 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.Limit(g(x),x=-1)=limit(g(x),x=-1);Il nous faut donc \303\251valuer les deux limites directionnelles.Limit(g(x),x=-1,left)=limit(g(x),x=-1,left);
Limit(g(x),x=-1,right)=limit(g(x),x=-1,right);Il y a donc une discontinuit\303\251 infinie en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYlLUY7Ni1RKCZtaW51cztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGWi1JI21uR0YkNiRRIjFGJ0Y+Rj5GK0Y+RisvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRkJGPg==. Ainsi, il y a une asymptote verticale d'\303\251quation 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.Esquissons le graphique de la fonction g pr\303\250s de cette asymptote verticale. Un trac\303\251 sur l'intervalle 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 devrait faire l'affaire. Contr\303\264lons l'affichage de l'axe des x et de
l'axe des y en pr\303\251cisant l'option view=[-3..3,-4..4].Asymptote_V:=plot([-1,y,y=-4..4],linestyle=3,thickness=0,color=khaki):
Graphe_2:=plot([x,g(x),x=-1.25..-.75],discont=true):
display({Graphe_2,Asymptote_V},view=[-3..3,-4..4]);La fonction g est un bel exemple qui nous montre que le calcul de la limite n'est pas n\303\251cessairement identique dans \342\204\202 et dans \342\204\235. \303\211valuons donc la limite \303\240 droite en
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 en cr\303\251ant, avec la macro-commande root, une fonction f d\303\251finie par 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.f:=x->root(x*(x-1)/(x^2-1),3);
Limit(f(x),x=-1,right)=limit(f(x),x=-1,right);C'est convaicant. Cet exemple souligne l'importance d'utiliser la macro-commande surd lorsqu'il s'agit d'analyser des fonctions r\303\251elles \303\240 valeurs r\303\251elles.
Pour faire l'\303\251tude des asymptotes horizontales, il nous faut \303\251valuer les limites pertinentes \303\240 moins l'infini et \303\240 plus l'infini.Limit(g(x),x=-infinity)=limit(g(x),x=-infinity);
Limit(g(x),x=+infinity)=limit(g(x),x=+infinity)On a donc une m\303\252me asymptote horizontale d'\303\251quation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ dans la partie n\303\251gative et positive de l'abscisse. Cr\303\251ons le trac\303\251 de l'asymptote horizontale et esquissons ensuite le trac\303\251 de la
fonction g pr\303\250s de cette asymptote.
Pour cr\303\251er ce trac\303\251, il faut consid\303\251rer des valeurs pour x de telles sortes que leur image soit assez pr\303\250s de 1 pour que le trac\303\251 pr\303\250s de l'asymptote horizontale soit correctement visible sur le graphique. Cr\303\251ons donc s\303\251par\303\251ment deux trac\303\251s de la fonction g: un premier sur l'intervalle 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 dans la partie n\303\251gative de l'abscisse et un second sur l'intervalle LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLUYvNiRRIjNGJ0YyRjJGMi8lJW9wZW5HUScmbHNxYjtGJy8lJmNsb3NlR1EnJnJzcWI7Ric= dans la partie positive de l'abscisse.
Ces intervalles vont faire l'affaire car ils ont \303\251t\303\251 trouv\303\251s apr\303\250s quelques essais. Il est assez rare d'obtenir, du premier coup, les meilleurs intervalles appropri\303\251s aux caract\303\251ristiques du trac\303\251 que l'on cherche \303\240 montrer. Attendez-vous qu'il soit souvent n\303\251cessaire de faire plusieurs essais.Asymptote_H:=plot([x,1,x=-3..3],linestyle=3,thickness=0,color=khaki):
Graphe_3:=plot([x,g(x),x=-3..-2]):
Graphe_4:=plot([x,g(x),x=2..3]):
display({Graphe_3,Graphe_4,Asymptote_H},view=[-3..3,-4..4]);Reste maintenant \303\240 superposer tous les \303\251l\303\251ments graphiques obtenus.display({Trou,Graphe_1,Graphe_2,Graphe_3,Graphe_4,
Asymptote_V,Asymptote_H},view=[-3..3,-4..4]);Il est n\303\251cessaire de poursuivre analytiquement l'\303\251tude d'une fonction pour que l'on puisse \303\252tre en mesure de bien r\303\251unir les morceaux que nous venons d'obtenir. C'est seulement apr\303\250s
l'introduction de la d\303\251riv\303\251e et de son interpr\303\251tation g\303\251om\303\251trique que nous serons en mesure de r\303\251unir correctement ces morceaux. Voici d'ailleurs une surprise si vous croyez que la r\303\251union des morceaux
\303\240 la droite de l'asymptote verticale se fait toujours en une courbe ouverte vers le bas. Ce n'est pas le cas.
En effet, cr\303\251ez directement le trac\303\251 de la fonction g sur l'intervalle 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. Superposons-y les asymptotes verticale et horizontale ainsi que le "trou''. Sp\303\251cifions l'option numpoints=200 pour un meilleur rendu de la courbure au voisinage de 0. Finalement, contr\303\264lons l'affichage des axes avec l'option view=[-3..3,-4..4].Graphe:=plot([x,g(x),x=-3..3],numpoints=200,discont=true):
display({Trou,Graphe,Asymptote_V,Asymptote_H},view=[-3..3,-4..4]);