[!!{tableau {matLi "%meta " "Valeur " } {matLi "%statut " "2 " } {matLi "%titre " "Utilisation de la croissance compar~00e9e pour calculer des limites ~005c(questionnaire~005c) " } {matLi "%auteur " "Essonnier Nataly " } {matLi "%langage " "fr " } {matLi "%description " "QRU d~00e9terminer le r~00e9sultat d~005c~0027une limite gr~00e2ce ~00e0 la croissance compar~00e9e " } {matLi "%discipline " "math~00e9matique " } {matLi "%theme " "fonctions " } {matLi "%age " " " } {matLi "%niveau " "BTS " } {matLi "%mots_cles " "limite, croissance compar~00e9e " } {matLi "%fiche_pedago " " " } {matLi "%licence " " " }}] " " [!!{header [{matLi [{tablefixeSSstop [{matLi {hide [noteMaxi:120]} {hide [nbDecimales:150]}}] [{matLi [{table00SS [{matLi {mText "Note"}}] [{matLi {mText "maximale"}}]}] [{table00SS [{matLi {mText "Nombre de"}}] [{matLi {mText "d~00e9cimales"}}]}]}] {matLi [20] [1]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [nom:120]} {hide [classe:120]}}] [{matLi {mText "Nom"} {mText "Classe"}}] [{matLi {mText " "} {mText " "}}]}]}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [nom:120]} {hide [classe:120]} {hide [correcteur:120]} {hide [points:120]} {hide [note:120]}}] [{matLi {mText "Nom"} {mText "Classe"} {mText "Examinateur"} {mText "Points"} {mText "Note"}}] [{matLi {mText " "} {mText " "} {mText "epsilon"} [?] [?]}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " "Cocher la bonne r~00e9ponse.[!AS;! ]" "Il ne peut y avoir qu~005c~0027une seule r~00e9ponse possible. " "Les r~00e9ponses erron~00e9es sont p~00e9nalis~00e9es.[!! ]" " " 1) " " {limit _[{right x +{infinity}}] {frac [x] [x]}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi 0 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } [{matLi " Explication[!! : ]" " " {limit _[{right x +{infinity}}] {frac [x] [x]}}={limit _[{right x +{infinity}}] 1}=1 " , on simplifie avant de faire le calcul de la limite car : " {limit _[{right x +{infinity}}] x}=+{infinity} " et " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne permet pas de conclure. " }]}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] {matLi " " }}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " 2) " " {limit _[{right x +{infinity}}] {frac {root [x]} [x]}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop {matLi 0 " " } [{matLi "Explication[!! : ]" " x croit beaucoup plus vite vers l~005c~0027infini que " {root [x]},donc " le rapport tend vers 0 d~005c~0027apr~00e8s le th~00e9or~00e8me de la croissance compar~00e9e. Mais nous pouvons ~00e9galement justifier ainsi : " {root [x]}*{root [x]}=x " , donc " {limit _[{right x +{infinity}}] {frac {root [x]} [x]}}={limit _[{right x +{infinity}}] {frac {root [x]} [{root [x]}*{root [x]}]}}={limit _[{right x +{infinity}}] {root [x]}}=+{infinity} " . " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] {root [x]}}=+{infinity} " et " {limit _[{right x +{infinity}}] x}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " }]}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] {matLi " " }}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " 3) " " {limit _[{right x +{infinity}}] {frac [x^[2]] [x]}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi 0 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] [{matLi " Explication[!! : ]" " x croit beaucoup moins vite vers l~005c~0027infini que " x^[2],donc " le rapport tend vers + " {infinity} " d~005c~0027apr~00e8s le th~00e9or~00e8me de la croissance compar~00e9e. Mais nous pouvons ~00e9galement justifier ainsi : " {limit _[{right x +{infinity}}] {frac [x^[2]] [x]}}={limit _[{right x +{infinity}}] x}=+{infinity} " . " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] {root [x]}}=+{infinity} " et " {limit _[{right x +{infinity}}] x}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " }]}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " 4) " " {limit _[{right x +{infinity}}] {frac [x^[2]] {root [x]}}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi 0 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] [{matLi " Explication[!! : ]" " " {root [x]} " croit beaucoup moins vite vers l~005c~0027infini que " x^[2],donc " le rapport tend vers + " {infinity}{prime d}apr~00e8s " le th~00e9or~00e8me sur la croissance compar~00e9e. Mais nous pouvons ~00e9galement justifier ainsi : " {limit _[{right x +{infinity}}] {frac [x^[2]] {root [x]}}}={limit _[{right x +{infinity}}] x*{root [x]}}=+{infinity} " . " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] {root [x]}}=+{infinity} " et " {limit _[{right x +{infinity}}] x^[2]}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " }]}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " 5) " " {limit _[{right x +{infinity}}] {frac [x] {ln (x)}}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi 0 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] [{matLi " [!BP!Explication :] " " x croit beaucoup plus vite vers l~005c~0027infini que " {ln (x)},donc " le rapport tend vers + " {infinity} " , c~005c~0027est l~005c~0027application du th~00e9or~00e8me sur la croissance compar~00e9e. Nous ne pouvons pas faire autrement. " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] {ln (x)}}=+{infinity} " et " {limit _[{right x +{infinity}}] x}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " }]}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " 6) " " {limit _[{right x +{infinity}}] {ln (x)}*e^[-x]}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop {matLi 0 " " } [{matLi "Explication [!!: ]" {limit _[{right x +{infinity}}] {ln (x)}}=+{infinity} " et " {limit _[{right x +{infinity}}] e^[-x]}=0 " or " +{infinity}*0 " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " "Mais nous pouvons ~00e9crire : " {limit _[{right x +{infinity}}] {ln (x)}*e^[-x]}={limit _[{right x +{infinity}}] {frac {ln (x)} [e^[x]]}}=0 " d~005c~0027apr~00e8s la croissance compar~00e9e, car " e^[x] " crit plus vite vers " +{infinity} " que ln~005c(x~005c). " }]}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] {matLi " " }}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " 7) " " {limit _[{right x +{infinity}}] {frac {ln (x)} [x]}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop {matLi 0 " " } [{matLi " " " Explication[!! : ]" " " {ln (x)} " croit beaucoup moins vite vers l~005c~0027infini que " x,donc " le rapport tend vers " 0 " d~005c~0027apr~00e8s le th~00e9or~00e8me sur la croissance compar~00e9e. " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] x}=+{infinity} " et " {limit _[{right x +{infinity}}] {ln (x)}}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " }]}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] {matLi " " }}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " " 8) " " {limit _[{right x +{infinity}}] {frac [e^[x]] [x^[2]]}}=? " " [!!{choice [{matLi [{tablefixeSSstop [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi 0 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtn}}]}] [{matLi [{hide {false}}]}]}] [{tablefixeSSstop {matLi " " 1 " " } {matLi " " }}]}] [{matLi [{table00SS [{matLi [{twoRadioBtn {radioBtnChecked}}]}] [{matLi [{hide {true}}]}]}] [{tablefixeSSstop [{matLi " " +{infinity} " " }] [{matLi " " " Explication[!! : ]" " " x^[2] " croit beaucoup moins vite vers l~005c~0027infini que " e^[x],donc " le rapport tend vers + " {infinity}{prime d}apr~00e8s " le th~00e9or~00e8me sur la croissance compar~00e9e. " " " "Remarque[!! : ]" {limit _[{right x +{infinity}}] e^[x]}=+{infinity} " et " {limit _[{right x +{infinity}}] x^[2]}=+{infinity} " mais " {frac [+{infinity}] [+{infinity}]} " est une forme ind~00e9termin~00e9e qui ne nous permet pas de conclure. " " " }]}]}]}]}] [{matLi [{tablefixeSSstop [{matLi {hide [scoreMax:120]} {hide [scoreMin:120]} {hide [coefIncorrect:380]}}] [{matLi [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "maximum"}}]} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}] [{table00SS [{matLi {mText "Score"}}] [{matLi {mText "minimum"}}]}] [{table00SS [{matLi {mText "Coefficient pour r~00e9ponse incorrecte"}}] [{matLi {mText "~005c(entre 1 et 4, peut ~00eatre d~00e9cimal~005c)"}}]}]}] {matLi [3] [0] [2]}}]}] [{matLi [{hide [{tablefixeSSstop [{matLi {hide [score:120]}}] [{matLi [{mText "Score"} {expMathEtInfo {unaryButton {mText "?"}} {nonExpress (commandeGenerale:edite:scoreExplication 0 jauneBouton 1 ~0027Score ~005c(explications~005c)~0027)}}]}] [{matLi {frac [3] [3]}}]}]}]}] [{matLi [{hide [{tablefixeSSstop {matLi "Annotation :" }}]}]}]}] " " " " " "