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Déterminer la limite en chacune des bornes de l'ensemble de définition d'une fonction f définie par f(x)=Q(lnx)Q est une fonction rationnelle    ressource 2735

Soit f la fonction définie pour tout x ] 0 ; 1 e 9 [ SequenceForm ] 0 ; -9 [ ] 1 e 9 ; e [ SequenceForm ] -9 ; [ ] e ; + [ SequenceForm ] ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", "0", "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", FractionBox["1", SuperscriptBox["\[ExponentialE]", "9"]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", 0, ";", Power[E, -9], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", FractionBox["1", SuperscriptBox["\[ExponentialE]", "9"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\[ExponentialE]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Power[E, -9], ";", E, "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", "\[ExponentialE]", "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", E, ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] 0 ; -9 [ SequenceForm ] -9 ; [ SequenceForm ] ; + DirectedInfinity 1 [ par f ( x ) = - 5 ( ln ( x ) + 1 ) ln 2 ( x ) + 8 ln ( x ) - 9 .

Calculez les limites de f aux bornes de son ensemble de définition.

- 0 +
lim x "\[Rule]" 0 x > 0 f ( x ) SequenceForm Underscript lim Underscript Rule x 0 x 0 f x =
lim x "\[Rule]" 1 e 9 x < 1 e 9 f ( x ) SequenceForm Underscript lim Underscript Rule x -9 x -9 f x =
lim x "\[Rule]" 1 e 9 x > 1 e 9 f ( x ) SequenceForm Underscript lim Underscript Rule x -9 x -9 f x =
lim x "\[Rule]" e x < e f ( x ) SequenceForm Underscript lim Underscript Rule x x f x =
lim x "\[Rule]" e x > e f ( x ) SequenceForm Underscript lim Underscript Rule x x f x =
lim x "\[Rule]" + f ( x ) SequenceForm Underscript lim Rule x pinf f x =