(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 9.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. 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Para ello necesitamos que f est\[EAcute] definida en \ puntos arbitrariamente pr\[OAcute]ximos a ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], ", es decir que haya puntos del ", Cell[BoxData[ FormBox[ RowBox[{"Dom", " ", "f"}], TraditionalForm]]], " a cualquier distancia de ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], ". No descartamos la posibilidad de el dominio de ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " incluya ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], ". Para que se cumplan estos objetivos ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], " debe ser un punto de acumulaci\[OAcute]n. " }], "Text", FontSize->18], Cell[TextData[{ "Podemos definir l\[IAcute]mite como sigue:\nSea f: D \[Subset] \ \[DoubleStruckCapitalR]\[RightArrow]", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "2"], TraditionalForm]]], " y sea ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], " un punto de acumulaci\[OAcute]n de D y sea un ", Cell[BoxData[ FormBox["l", TraditionalForm]]], " \[Element] \[DoubleStruckCapitalR]. Se dice que ", Cell[BoxData[ UnderscriptBox["lim", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Rule]", RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}]}]]], FontSize->18], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"x", ",", "y"}], " ", ")"}], "=", " ", "l"}], TraditionalForm]]], ", si ", Cell[BoxData[ FormBox[ RowBox[{"\[ForAll]", RowBox[{"\[Epsilon]", ">", "0", " "}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"\[Exists]", RowBox[{"\[Delta]", ">", "0", " "}]}], TraditionalForm]]], " tal que ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], "\[Element]D,", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}], " ", "-", "l"}], "|", RowBox[{"<", "\[Epsilon]", " "}]}], TraditionalForm]]], " siempre que ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "0"]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"y", "-", SubscriptBox["y", "0"]}], ")"}], "2"]}]], "<", "\[Delta]"}], TraditionalForm]]] }], "Text", FontSize->18], Cell[TextData[{ "Ejercicio.- Generalicese la definici\[OAcute]n de limite a ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " varibles en ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], ". 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Para calcular los l\[IAcute]mites seg\[UAcute]n la direcci\[OAcute]n de \ aproximaci\[OAcute]n a ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], " podemos utilizar la siguiente proposici\[OAcute]n.\nSi existe ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", ">", "0", " "}], TraditionalForm]]], "tal que todos los puntos de ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"B", "(", RowBox[{"a", ",", "\[Epsilon]"}], ")"}], " "}], TraditionalForm]]], ", con ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], ", pertenecientes a D [Observaci\[OAcute]n: ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], ")"}], TraditionalForm]]], " puede no pertenecer a D] y ", Cell[BoxData[ FormBox[ RowBox[{"l", "=", RowBox[{ UnderscriptBox["lim", 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m\[AAcute]ximo" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "/.", RowBox[{"{", RowBox[{"\[Lambda]", "->", RowBox[{"-", FractionBox["5", "2"]}]}], "}"}], " ", RowBox[{"(*", "Maximo", "*)"}]}]], "Input"], Cell[BoxData[ RowBox[{"-", "5"}]], "Output"] }, Open ]], Cell["El m\[AAcute]ximo est\[AAcute] en", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"z", "[", RowBox[{ RowBox[{"-", FractionBox["4", "5"]}], ",", " ", RowBox[{"-", FractionBox["3", "5"]}]}], "]"}]], "Input"], Cell[BoxData["11"], "Output"] }, Open ]], Cell[TextData[{ "Para \[Lambda] = ", Cell[BoxData[ FractionBox["5", "2"]]], " \[CapitalDelta] >0 , y A>0, luego hay un m\[IAcute]nimo" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "/.", RowBox[{"{", RowBox[{"\[Lambda]", "->", FractionBox["5", "2"]}], "}"}], RowBox[{"(*", "M\[IAcute]nimo", "*)"}]}]], "Input"], Cell[BoxData["5"], "Output"] }, Open ]], Cell["El m\[AAcute]ximo est\[AAcute] en", "Text"], Cell[CellGroupData[{ Cell[BoxData[ 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