[!!{tableau {matLi "%meta " "Valeur " } {matLi "%statut " "2 " } {matLi "%titre " "Cours sur les limites et asymptotes " } {matLi "%auteur " "Essonnier Nataly" } {matLi "%langage " "fr " } {matLi "%description " "Interpr~00e9ter une limite en terme d~005c~0027asymptote verticale, horizontale ou comment montrer qu~005c~0027une droite est asymptote oblique " } {matLi "%discipline " "Math~00e9matique " } {matLi "%theme " "Asymptote " } {matLi "%age " " " } {matLi "%niveau " "BTS " } {matLi "%mots_cles " "Asymptote " } {matLi "%fiche_pedago " " " } {matLi "%licence " " " }}] " " "D~00e9terminer une asymptote verticale, horizontale ou oblique. " " " " " "Soit [!BP!C] la [!BP!courbe repr~00e9sentative] de la fonction[!BP! f d~00e9finie ]sur un intervalle " [!%format:normal;align:left;!{include {include I Df} {real}}] " . " [!%format:normal;align:left;!Df] " : ensemble de d~00e9finition de la fonction " [!CP!f] " . " " " "Nous ne nous int~00e9ressons qu~005c~0027~00e0[!BP! trois types] d~005c~0027asymptotes : [!BP!verticales], [!BP!horizontales] et [!BP!obliques]. " " " " " "Technique : [!! Nous utilisons le calcul de limites pour d~00e9terminer une asymptote. ]" " " "Verticale :[!AS! ]" "Soit un nombre r~00e9el a. " "Si " {limit _{right x a} f(x)}=+{infinity} " ou- " {infinity} " alors " x=a " est l~005c~0027~00e9quation cart~00e9sienne de [!BP!l~005c~0027asymptote verticale] ~00e0 [!BP!C] . " "Exemple :[!! ]" "Soit " f(x)={frac [1] [(x-2)^[2]]} " sur " {OOinterval [2;+{infinity}]} " . " [!%align:left;!{limit _{right x 2} f(x)}=+{infinity}] " alors " [!%align:left;!x=2] " est l~005c~0027~00e9quation cart~00e9sienne de [!BP!l~005c~0027asymptote verticale] ~00e0 [!BP!C]. " " " " " "Horizontale :[!! ]" "Si " {limit _[{natOr {right x +{infinity}} -{infinity}}] f(x)}=b " , " {member b {real}} " alors " y=b " est l~005c~0027~00e9quation cart~00e9sienne de [!BP!l~005c~0027asymptote horizontale] ~00e0 [!BP!C] en " +{infinity} " ,respectivement en " -{infinity} " . " " " "Exemple :[!! ]" "Soit " f(x)=6-{frac [5] [x+1]} " sur " {OOinterval [-{infinity};-1]} " . " {limit _[{right x -{infinity}}] f(x)}=6 " alors " y=6 " est l~005c~0027~00e9quation cart~00e9sienne de [!BP!l~005c~0027asymptote horizontale] ~00e0 [!BP!C] en " -{infinity} " . " " " " " "Oblique :[!! ]" "Soit la droite D d~005c~0027~00e9quation cart~00e9sienne " y=ax+b " . " "Si " {limit _[{natOr {right x +{infinity}} -{infinity}}] (f(x)-y)}=0 " alors " y=ax+b " est l~005c~0027~00e9quation cart~00e9sienne de [!BP!l~005c~0027asymptote oblique] ~00e0 [!BP!C] en " +{infinity} " ,respectivement en " -{infinity} " .[!BP! ]" "Nous pouvons dire que D est asymptote oblique ~00e0 C en " +{infinity} " , respectivement en " -{infinity} " . " " " "Exemple :[!! ]" "Soient " f(x)=3x-2+{frac [3] [1-2x]} " sur " {OOinterval [{frac [1] [2]};+{infinity}]} " et D: " y=3x-2 " . " {limit _[{right x +{infinity}}] (f(x)-y)}={limit _[{right x +{infinity}}] {frac [3] [1-2x]}}=0 " alors D est asymptote oblique ~00e0 C en " +{infinity} " . " " " " " "Remarque[!! : vous pouvez utiliser ][!BP!g~00e9og~00e9bra][!! pour tracer la courbe C et son asymptote D] [!!pour ensuite comparer le comportement graphique des deux courbes. ]" " "