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Déterminer la limite à l'infini d'une fonction rationnelle    ressource 1420

Soit f la fonction définie pour tout x ] - ; - 4 [ SequenceForm ] DirectedInfinity -1 ; -4 [ ] - 4 ; + [ SequenceForm ] -4 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", -4, "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", -4, ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -4 [ SequenceForm ] -4 ; + DirectedInfinity 1 [ par f ( x ) = - x 2 - 6 x - 10 x + 4 .

On admet que, pour tout x ] - ; - 4 [ SequenceForm ] DirectedInfinity -1 ; -4 [ ] - 4 ; + [ SequenceForm ] -4 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", -4, "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", -4, ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -4 [ SequenceForm ] -4 ; + DirectedInfinity 1 [ , f ( x ) = - x + 2 - x - 4 - 2 .

On se propose d'étudier la limite de f en + .

Soit g la fonction définie pour tout x ] - ; - 4 [ SequenceForm ] DirectedInfinity -1 ; -4 [ ] - 4 ; + [ SequenceForm ] -4 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", -4, "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "4"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", -4, ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -4 [ SequenceForm ] -4 ; + DirectedInfinity 1 [ par g ( x ) = - x - 4 .

Déterminez la limite de g en + .

lim x "\[Rule]" + g ( x ) =