(* 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. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1063, 20] NotebookDataLength[ 334938, 8950] NotebookOptionsPosition[ 308590, 8091] NotebookOutlinePosition[ 314523, 8272] CellTagsIndexPosition[ 314453, 8267] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Funciones, l\[IAcute]mites y continuidad. \ \>", "Title", CellChangeTimes->{{3.402221900287321*^9, 3.402221926909873*^9}, { 3.40275716428125*^9, 3.402757177328125*^9}, {3.403330700391478*^9, 3.403330706124502*^9}, 3.404975418375*^9, 3.5581919694451027`*^9, { 3.5591301646762185`*^9, 3.559130170822629*^9}, {3.559738654177397*^9, 3.559738664598298*^9}, {3.5610188519822817`*^9, 3.5610188697506995`*^9}, 3.561213900127493*^9}], Cell[CellGroupData[{ Cell[TextData[{ "Guillermo S\[AAcute]nchez (", ButtonBox["http://diarium.usal.es/guillermo", BaseStyle->"Hyperlink", ButtonData->{ URL["http://diarium.usal.es/guillermo"], None}, ButtonNote->"http://diarium.usal.es/guillermo"], ")" }], "ItemParagraph", CellChangeTimes->{{3.4817991669102297`*^9, 3.48179917634823*^9}, { 3.50652102881168*^9, 3.5065210329479165`*^9}, {3.506616173311325*^9, 3.506616182905342*^9}, {3.50661660323248*^9, 3.506616628130124*^9}, { 3.5096080657618456`*^9, 3.5096080703794537`*^9}, {3.5758122954985123`*^9, 3.575812327640621*^9}}], Cell["\<\ Departamento de Economia e H\.aa Econ\[OAcute]mica. 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En su lugar preferiamos utilizar el estilo ", Cell[BoxData[ ButtonBox["TraditionalForm", BaseStyle->"Link", ButtonData->"paclet:ref/TraditionalForm"]]], " que da una apariencia a las salidas (", StyleBox["Output", FontSlant->"Italic"], ") coincidente con el habitualmente utilizado en la notaci\[OAcute]n cl\ \[AAcute]sica utilizada en las matem\[AAcute]ticas. 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En el caso particular que corresponda a cada elemento de A \ corresponda un s\[OAcute]lo elemento de B dedimos que la funci\[OAcute]n es \ inyectiva (ej.: A cada contribuyente debe corresponder un s\[OAcute]lo NIF) \ \>", "Text", FontSize->16], Cell[TextData[{ "Las funciones reales de variable real pueden representarse \ gr\[AAcute]ficamente en el plano XY, en X (eje horizontal o de abcisas) se \ representa ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " y en Y (eje vertical o de ordenadas) ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]]], "." }], "Text", FontSize->16] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Dominio. Calcular el dominio para las funciones que se indican\ \>", "Section", CellChangeTimes->{{3.559737927140563*^9, 3.559737965392203*^9}, { 3.559739029657426*^9, 3.559739037254702*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Las funciones polinomicas tiene como dominio ", Cell[BoxData[ FormBox[Cell["\[DoubleStruckCapitalR]"], TraditionalForm]]], " . 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excepto para Q(x)=0,\ \>", "Subsection", CellChangeTimes->{{3.5609337763222065`*^9, 3.560933795900402*^9}, { 3.5609338400644846`*^9, 3.560933911013986*^9}}, FontSize->18], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"Ejemplo", ":", " ", "y"}], " ", "=", FractionBox[ RowBox[{"x", "-", "2"}], RowBox[{"x", "-", "5"}]], " "}]}]], "Subsubsection", Evaluatable->False, CellChangeTimes->{{3.56093372663568*^9, 3.5609337358397455`*^9}, 3.560933800034444*^9, {3.5609339828371305`*^9, 3.560933986799569*^9}}, FontSize->18], Cell[TextData[{ "Su dominio es \[DoubleStruckCapitalR]-{5}, porque su cociente est\[AAcute] \ definido para cualquier valor de \[DoubleStruckCapitalR] - 5 (porque para x \ =5 se hace cero el denominador y por tanto ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " toma un valor infinito) " }], "Text", CellChangeTimes->{{3.560316485712271*^9, 3.5603165561624837`*^9}, { 3.5603169800344543`*^9, 3.5603169848706193`*^9}, {3.560608853829282*^9, 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0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, PlotRange->{{-2, 2}, {-0.9999996208907248, 14.999998693877577`}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}], TraditionalForm]], "Output", CellChangeTimes->{3.5612926893394375`*^9}] }, Open ]], Cell["\<\ \[DownQuestion]Que valor toma la siguiente funci\[OAcute]n en x = 2?\ \>", "Text", CellChangeTimes->{{3.561292995897812*^9, 3.561293028595674*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{" ", RowBox[{ SuperscriptBox["x", "2"], "-", "1"}]}], RowBox[{"x", "-", "1"}]], "==", " ", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}], RowBox[{"x", "-", "1"}]]}]], "Input", CellChangeTimes->{{3.561292904012943*^9, 3.561292979704813*^9}}], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", "1"}], RowBox[{"x", "-", "1"}]], "\[LongEqual]", RowBox[{"x", "+", "1"}]}], TraditionalForm]], "Output", CellChangeTimes->{{3.5612929078037686`*^9, 3.56129292123549*^9}, 3.5612929848372927`*^9}] }, Open ]], Cell["\<\ Definici\[OAcute]n de l\[IAcute]mite (no rigurosa) es la siguiente:\ \>", "Text", FontSize->18], Cell[TextData[{ Cell[BoxData[ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", SubscriptBox["x", "0"]}]]], FontSize->18], " ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]]], " = A quiere decir ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]]], " se puede hacer tan pr\[OAcute]ximo a L como queramos, para todo ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " pr\[OAcute]ximo (pero no igual) a ", Cell[BoxData[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]]]]], FontSize->18], ". " }], "Text", CellFrame->{{3, 0}, {0, 0.5}}, CellChangeTimes->{{3.561292337337849*^9, 3.5612923395374312`*^9}, { 3.561292427693924*^9, 3.5612924292226458`*^9}}, FontSize->18], Cell["\<\ La pr\[OAcute]ximidad, o mas generalmente la distancia, en \ \[DoubleStruckCapitalR] , es el valor absoluto de la diferencia entre ellos.\ \>", "Text", CellChangeTimes->{{3.5597390719806337`*^9, 3.5597390947412586`*^9}}, FontSize->18], Cell["\<\ \[DownQuestion]Que n\[UAcute]meros x distan de 5 menos de 0.1?\ \>", "Text", CellChangeTimes->{{3.5597390719806337`*^9, 3.559739125660759*^9}, { 3.560609061950836*^9, 3.560609069719715*^9}, 3.5606091203110466`*^9}, FontSize->18], Cell[TextData[{ "Resp.: ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"|", RowBox[{"x", " ", "-", "5"}], "|"}]}], TraditionalForm]]], "\[Precedes] 0.1, que es equivalente a ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", " ", "0.1"}], "\[Precedes]", RowBox[{"x", " ", "-", "5"}], "\[Precedes]", "0.1"}], TraditionalForm]]], " o ", Cell[BoxData[ FormBox[ RowBox[{"4.9", "\[Precedes]", "x", "\[Precedes]", "5.1"}], TraditionalForm]]], ". Es decir, los n\[UAcute]meros que distan de 5 menos de 0.1 son los \ comprendidos entre 5.1 y 4.9.\n" }], "ItemParagraph", CellChangeTimes->{{3.5597390719806337`*^9, 3.559739125660759*^9}, { 3.560609061950836*^9, 3.560609069719715*^9}, {3.560950117708661*^9, 3.560950121707123*^9}}, FontSize->18], Cell[TextData[{ "\[DownQuestion]Que n\[UAcute]meros ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " distan de ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], " menos de \[Delta]?" }], "Text", CellChangeTimes->{{3.5597390719806337`*^9, 3.559739100560115*^9}, { 3.560609115755763*^9, 3.5606091235714917`*^9}}, FontSize->18], Cell[TextData[{ "Resp.: ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"|", RowBox[{"x", " ", "-", SubscriptBox["x", "0"]}], "|"}]}], TraditionalForm]]], "\[Precedes] \[Delta], que es equivalente a ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "\[Delta]"}], "\[Precedes]", RowBox[{"x", " ", "-", SubscriptBox["x", "0"]}], "\[Precedes]", "\[Delta]"}], TraditionalForm]]], " o ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["x", "0"], "-", "\[Delta]"}], "\[Precedes]", "x", " ", "\[Precedes]", RowBox[{ SubscriptBox["x", "0"], "+", "\[Delta]"}]}], TraditionalForm]]], ", que tambien se puede escribir: x \[Element] (", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["x", "0"], "-", "\[Delta]"}], ",", " ", RowBox[{ SubscriptBox["x", "0"], "+", "\[Delta]"}]}], TraditionalForm]]], ")" }], "ItemParagraph", CellChangeTimes->{{3.5597390719806337`*^9, 3.559739100560115*^9}, 3.560609115755763*^9}, FontSize->18], Cell["\<\ Podemos ahora reformular la definici\[OAcute]n de l\[IAcute]mite como sigue:\ \>", "Text", CellChangeTimes->{ 3.5597391082665896`*^9, {3.560317032357486*^9, 3.56031704380793*^9}}, FontSize->18], Cell[TextData[{ Cell[BoxData[ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", SubscriptBox["x", "0"]}]]], FontSize->18], " ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]]], " = ", StyleBox["L", FontSlant->"Italic"], " quiere decir que podemos hacer", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"|", RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "-", "L"}], "|"}]}], TraditionalForm]]], " tan peque\[NTilde]o como queramos \[ForAll] ", Cell[BoxData[ FormBox[ RowBox[{"x", " ", "\[NotEqual]", FormBox[ SubscriptBox["x", "0"], TraditionalForm]}], TraditionalForm]]], " con ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"x", " ", "-", " ", SubscriptBox["x", "0"]}], " ", "|"}], TraditionalForm]]], " suficientemente peque\[NTilde]o. (Observese que", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"x", " ", "\[NotEqual]", FormBox[ SubscriptBox["x", "0"], TraditionalForm]}]}], TraditionalForm]]], " es equivalente a ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0", "\[Precedes]"}], "|", RowBox[{"x", " ", "-", " ", SubscriptBox["x", "0"]}], " ", "|"}], TraditionalForm]]], " ) " }], "Text", CellFrame->{{3, 0}, {0, 0.5}}, CellChangeTimes->{ 3.5597391082665896`*^9, {3.560317032357486*^9, 3.56031704380793*^9}, { 3.5612924450723886`*^9, 3.561292452373255*^9}}, FontSize->18], Cell["\<\ La definici\[OAcute]n de l\[IAcute]mite puese a\[UAcute]n precisarse m\ \[AAcute]s expres\[AAcute]ndola en t\[EAcute]rminos de \[CurlyEpsilon] y \ \[Delta] (que es definici\[OAcute]n m\[AAcute]s utilizada).\ \>", "Text", FontSize->18], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], " "}], TraditionalForm]]], "tiende al l\[IAcute]mite L cuando ", Cell[BoxData[ FormBox[ RowBox[{"x", " "}], TraditionalForm]]], "tiende a ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], " , y escribimos ", Cell[BoxData[ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", SubscriptBox["x", "0"]}]]], FontSize->18], " ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]]], " = ", StyleBox["L", FontSlant->"Italic"], ", si ", Cell[BoxData[ FormBox[ RowBox[{"\[ForAll]", RowBox[{"\[Epsilon]", ">", "0", " "}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{"\[Exists]", RowBox[{"\[Delta]", ">", "0", " "}]}], TraditionalForm]]], " tal que ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "-", RowBox[{"f", "(", SubscriptBox["x", "0"], ")"}]}], "|", RowBox[{"<", "\[Epsilon]", " "}]}], TraditionalForm]]], " siempre que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0", "<"}], "|", RowBox[{"x", "-", SubscriptBox["x", "0"]}], "|", RowBox[{"<", "\[Delta]"}]}], TraditionalForm]]] }], "Text", CellFrame->{{3, 0}, {0, 0.5}}, CellChangeTimes->{{3.5612924624041457`*^9, 3.561292469751867*^9}}, FontSize->18], Cell[TextData[{ "Ejemplo: el ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "3"}]], RowBox[{"(", RowBox[{ RowBox[{"3", " ", "x"}], "-", "2"}], ")"}]}], "=", "7"}], TraditionalForm]]], " utilizando la definici\[OAcute]n de l\[IAcute]mite se calcula como sigue." }], "Text", CellChangeTimes->{{3.5600555188910894`*^9, 3.560055535629213*^9}, { 3.560056154348956*^9, 3.5600562062332525`*^9}, {3.560317555789689*^9, 3.5603175912488737`*^9}, {3.5603176318715987`*^9, 3.5603176365985413`*^9}, 3.560317791305318*^9, {3.5758124184032817`*^9, 3.575812420626581*^9}, { 3.575812653915383*^9, 3.5758126770673585`*^9}, 3.575812717490638*^9}, FontSize->18], Cell[TextData[{ "Sol.: \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "=", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", "x"}], "-", "2"}], ")"}]}], ",", " ", RowBox[{"a", "=", "2"}], ",", " ", RowBox[{"A", " ", "=", " ", "7"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"|", RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "-", "A"}], "|"}], " ", "=", " ", RowBox[{ RowBox[{"|", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"3", " ", "x"}], "-", "2"}], ")"}], " ", "-", " ", "7"}], "|"}], " ", "=", " ", RowBox[{ RowBox[{"|", RowBox[{ RowBox[{"3", " ", "x"}], " ", "-", " ", "9"}], "|"}], " ", "=", " ", RowBox[{"3", "|", RowBox[{"x", "-", "3"}], "|", RowBox[{"<", "\[Epsilon]"}]}]}]}]}], TraditionalForm]]], " que verifica siempre ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0", "<"}], "|", RowBox[{"x", "-", "3"}], "|", RowBox[{"<", RowBox[{"\[Epsilon]", "/", "3"}]}]}], TraditionalForm]]], ". \nPor tanto:\n", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "-", "A"}], "|", " ", RowBox[{"<", " ", "\[Epsilon]", " "}]}], TraditionalForm]]], "si ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"x", "-", "3"}], "|", RowBox[{"<", "\[Delta]", " "}]}], TraditionalForm]]], ", que se verifica para cualquier valor \[Delta]\[LessSlantEqual] \ \[Epsilon]/3" }], "Text", FontSize->18], Cell[TextData[{ "Ejemplo: Comprobar que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "a"}]], SqrtBox["x"]}], "=", SqrtBox["a"]}], TraditionalForm]]], " utilizando la definici\[OAcute]n de l\[IAcute]mite. " }], "Text", CellChangeTimes->{{3.5600555659641314`*^9, 3.560055567370696*^9}, { 3.560056210437131*^9, 3.56005621718834*^9}, {3.560317602309368*^9, 3.5603176089082556`*^9}, {3.5603176414501324`*^9, 3.5603176504670215`*^9}, { 3.575812423602892*^9, 3.5758124254431477`*^9}, {3.575812704058793*^9, 3.575812732746519*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"Sol", "."}], ":", "\[IndentingNewLine]", RowBox[{"|", RowBox[{ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "-", "A"}], "|"}], " ", "=", " ", RowBox[{"|", RowBox[{ FormBox[ SqrtBox["x"], TraditionalForm], "-", FormBox[ SqrtBox["a"], TraditionalForm]}], "|", RowBox[{"<", "\[Epsilon]"}]}]}], TraditionalForm], " ", "cuando", RowBox[{ FormBox[ RowBox[{ RowBox[{"0", "<"}], "|", RowBox[{"x", "-", "a"}], "|", RowBox[{"<", "\[Delta]"}]}], TraditionalForm], "."}]}]}]}]], "Text", FontSize->18], Cell["En este caso hacemos:", "Text", FontSize->18], Cell[BoxData[{ RowBox[{ RowBox[{"|", RowBox[{ FormBox[ 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Esto se conoce como indeterminaci\[OAcute]n pues no se puede asegurar \ nada sobre el l\[IAcute]mite de la funcion en ese punto. En ese caso hay que \ proceder a resolver la indeterminaci\[OAcute]n." }], "Text", CellChangeTimes->{{3.561051170873352*^9, 3.5610511938802376`*^9}, { 3.561051410631421*^9, 3.5610514108474455`*^9}, 3.5610514465029173`*^9, { 3.5610514899593678`*^9, 3.56105157176563*^9}, {3.5610516021124363`*^9, 3.561052076342907*^9}, {3.561052190127181*^9, 3.5610522006144958`*^9}, { 3.561052230672266*^9, 3.5610522445300045`*^9}, {3.5610522890125785`*^9, 3.561052298755805*^9}, {3.561214289485094*^9, 3.5612143162713356`*^9}}], Cell[TextData[{ "Por ejemplo: ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", RowBox[{"-", FractionBox["1", "2"]}]}]], "\[ThinSpace]", FractionBox[ RowBox[{ RowBox[{"4", " ", SuperscriptBox["x", "2"]}], "-", "1"}], RowBox[{ RowBox[{"4", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"8", " ", "x"}], "+", "3"}]]}], TraditionalForm]], CellChangeTimes->{3.5603181995459433`*^9, 3.561052223730398*^9}], "= ", " ", Cell[BoxData[ FormBox[ FractionBox["0", "0"], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.561051170873352*^9, 3.5610511938802376`*^9}, { 3.561051410631421*^9, 3.5610514108474455`*^9}, 3.5610514465029173`*^9, { 3.5610514899593678`*^9, 3.56105157176563*^9}, {3.5610516021124363`*^9, 3.561052076342907*^9}, {3.561052190127181*^9, 3.5610522006144958`*^9}, { 3.561052230672266*^9, 3.561052260368992*^9}}], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", RowBox[{"-", FractionBox["1", "2"]}]}]], "\[ThinSpace]", FractionBox[ RowBox[{ RowBox[{"4", " ", SuperscriptBox["x", "2"]}], "-", "1"}], RowBox[{ RowBox[{"4", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"8", " ", "x"}], "+", "3"}]]}], TraditionalForm]], "Input", CellChangeTimes->{3.5603181995459433`*^9, 3.561052223730398*^9}, CellTags->"Limit"], Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]], "Output", CellChangeTimes->{3.559738698169819*^9}, CellTags->"Limit"] }, Open ]], Cell[TextData[{ "Para una descripci\[OAcute]n detallada de como se resuelven los distintos \ tipos de inderminaciones puede consultar: ", ButtonBox["http://www.unizar.es/aragon_tres/unidad7/u7fun/u7funte30.pdf", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.unizar.es/aragon_tres/unidad7/u7fun/u7funte30.pdf"], None}, ButtonNote->"http://www.unizar.es/aragon_tres/unidad7/u7fun/u7funte30.pdf"] }], "Text", CellChangeTimes->{{3.561121569107445*^9, 3.5611216578910775`*^9}}], Cell["\<\ El l\[IAcute]mite siguiente es una indeterminaci\[OAcute]n de la forma 0/0.\ \>", "Text", CellChangeTimes->{{3.561399678763857*^9, 3.5613997341751537`*^9}, { 3.5613998258409147`*^9, 3.5613998523921614`*^9}}], Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "3"}]], "\[ThinSpace]", FractionBox[ RowBox[{ SuperscriptBox["x", "3"], "-", "27"}], RowBox[{ SuperscriptBox["x", "2"], "-", "9"}]]}], TraditionalForm]], "Input", CellChangeTimes->{{3.5613833760766306`*^9, 3.5613834410670643`*^9}}], Cell["\<\ En estos casos (numerador y denominador polinomios) se resuelven factorizando \ y simplificando, como se muestra\ \>", "Text", CellChangeTimes->{{3.561399861892578*^9, 3.5613999389567137`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"lim", FractionBox[ RowBox[{ SuperscriptBox["x", "3"], "-", "27"}], RowBox[{ SuperscriptBox["x", "2"], "-", "9"}]]}], "\[Equal]", " ", RowBox[{"lim", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"3", " ", "x"}], "+", "9"}], ")"}]}], RowBox[{ RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "3"}], ")"}]}]]}]}], "=", FractionBox["9", "2"]}]], "Text", CellChangeTimes->{{3.5613836165532365`*^9, 3.5613837264743595`*^9}, 3.561386686721893*^9, {3.5613885401682234`*^9, 3.5613885469188986`*^9}, { 3.5613998557617674`*^9, 3.56139985713457*^9}}], Cell["En ejemplos como el siguiente: ", "Text", CellChangeTimes->{{3.5613999586751485`*^9, 3.561400024429264*^9}}], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "4"}]], "\[ThinSpace]", FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", "16"}], RowBox[{"2", "-", SqrtBox["x"]}]]}], TraditionalForm]], "Input", CellTags->"Limit"], Cell[BoxData[ FormBox[ RowBox[{"-", "32"}], TraditionalForm]], "Output", CellChangeTimes->{3.5597386983726225`*^9}, CellTags->"Limit"] }, Open ]], Cell["\<\ El denominador incluye una raiz. Se pueden resolver multiplicando el \ denominador por el conjugado\ \>", "Text", CellChangeTimes->{{3.5613832694680552`*^9, 3.561383279920189*^9}, { 3.5614000306848745`*^9, 3.561400061416929*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"lim", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "4"}], ")"}], " ", RowBox[{"(", RowBox[{"x", "+", "4"}], ")"}], RowBox[{"(", RowBox[{"2", "+", SqrtBox["x"]}], ")"}]}], RowBox[{ RowBox[{"(", RowBox[{"2", "-", SqrtBox["x"]}], ")"}], RowBox[{"(", RowBox[{"2", "+", SqrtBox["x"]}], ")"}]}]]}], "=", RowBox[{ RowBox[{"lim", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "4"}], ")"}], " ", RowBox[{"(", RowBox[{"x", "+", "4"}], ")"}], RowBox[{"(", RowBox[{"2", "+", SqrtBox["x"]}], ")"}]}], RowBox[{"-", RowBox[{"(", RowBox[{"x", "-", "4"}], ")"}]}]]}], "=", RowBox[{ RowBox[{"lim", "(", RowBox[{ RowBox[{"-", " ", RowBox[{"(", RowBox[{"x", "+", "4"}], ")"}]}], RowBox[{"(", RowBox[{"2", "+", SqrtBox["x"]}], ")"}]}], ")"}], "=", RowBox[{"-", 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Tiene una \ discontinuidad de primera especie . 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" }], "Text", CellChangeTimes->{{3.5612147778631587`*^9, 3.5612148759427943`*^9}}, FontSize->18] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2.- Demostrar que la funci\[OAcute]n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "=", " ", RowBox[{"|", "x", "|", " "}]}]}], TraditionalForm]]], "es continua en R." }], "Subsubsection", CellChangeTimes->{{3.5603178653281*^9, 3.560317897246025*^9}}, FontSize->18], Cell["\<\ Nota: Para probar que una funci\[OAcute]n es continua se puede utilizar \ tambien el siguiente criterio: \ \>", "Text", FontSize->18], Cell[TextData[{ "En ", Cell[BoxData[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", SubscriptBox["x", "0"]}]], "\[ThinSpace]", RowBox[{"f", RowBox[{"(", "x", ")"}]}]}], " ", "=", " ", RowBox[{"f", RowBox[{"(", SubscriptBox["x", "0"], ")"}]}]}]], "DisplayFormula", FontSize->16, CellTags->"Limit"], "se hace la sustituci\[OAcute]n: ", Cell[BoxData[ RowBox[{"x", "=", RowBox[{ 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Entonces f(x) toma valores intermedios entre f(a) y f(b) cuando x recorre \ [a, b] " }], "Text", FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["Teorema de Bolzano", "Subsection", CellChangeTimes->{{3.5609519091476393`*^9, 3.560951915019348*^9}}], Cell["\<\ Sea f una funci\[OAcute]n continua en [a, b] tal que f(a) y f(b) tienen \ signos distintos, entonces existe al menos un c\[Element](a, b) tal que f(c) \ = 0\ \>", "Text", CellChangeTimes->{{3.5609518025010796`*^9, 3.5609518042913485`*^9}, { 3.560951917347681*^9, 3.5609519209000807`*^9}}, FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["Teorema de los valores extremos", "Subsection", CellChangeTimes->{{3.5609529409203873`*^9, 3.560952942799571*^9}}], Cell["\<\ Una de las aplicaciones mas importantes del an\[AAcute]lisis, en particular \ en el an\[AAcute]lisis econ\[OAcute]mico, es el c\[AAcute]lculo de \ m\[AAcute]ximos y m\[IAcute]nimos conocidos como puntos \[OAcute]ptimos. As\ \[IAcute] si D es el dominio de f(x) entonces\ \>", "Text", FontSize->16], Cell["\<\ c \[Element] D es un m\[AAcute]ximo de f \[DoubleLongLeftRightArrow] f(x) \ \[LessEqual] f(c) \[ForAll] x \[Element] D \ \>", "Text", FontSize->16], Cell["\<\ d \[Element] D es un m\[IAcute]nimo de f \[DoubleLongLeftRightArrow] f(x) \ \[GreaterEqual] f(d) \[ForAll] x \[Element] D \ \>", "Text", FontSize->16], Cell[TextData[{ StyleBox["Teorema", FontWeight->"Bold"], ": Si f una funci\[OAcute]n continua en el intervalo cerrado [a, b] cerrado \ y acotado, tiene en \[EAcute]l un m\[AAcute]ximo y un m\[IAcute]nimo." }], "Text", FontSize->16], Cell["\<\ Obs\[EAcute]rvese que se trata de una condici\[OAcute]n suficiente pero no \ necesaria \ \>", "Text", FontSize->16], Cell[TextData[{ StyleBox["Teorema", FontWeight->"Bold"], ": Supongamos que f est\[AAcute] definida en el intervalo I y sea c un \ punto interior de I (esto es distinto del punto inicial y final. 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Segun el \ teorema de Bolzano habra un valor intermedio x=c para el que f(c)=0 (aunque \ tambien es evidente graficamente), que ser\[AAcute] la solucion. Podemos \ proseguir hasta encontrar una difencia entre suficientemente peque\[NTilde]a.\ \>", "Text", CellChangeTimes->{{3.5609528890457525`*^9, 3.5609529028185263`*^9}, { 3.560953022858772*^9, 3.5609533152838235`*^9}, {3.560953382386281*^9, 3.5609534514750075`*^9}, 3.560953822499059*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"1", " ", "+", " ", RowBox[{"Sin", "[", "2.5", "]"}], "-", SqrtBox["2.5"]}]], "Input", CellChangeTimes->{{3.5609529502545414`*^9, 3.5609529577575006`*^9}, { 3.5609530870778527`*^9, 3.5609530910613832`*^9}, {3.560953338334675*^9, 3.560953341717147*^9}, {3.560953803374595*^9, 3.560953847926284*^9}}], Cell[BoxData[ FormBox["0.07251711788465087`", TraditionalForm]], "Output", CellChangeTimes->{3.5609529585645905`*^9, 3.560953095651922*^9, 3.5609533699226823`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"1", " ", "+", " ", RowBox[{"Sin", "[", "2.55", "]"}], "-", SqrtBox["2.55"]}]], "Input", CellChangeTimes->{{3.5609529502545414`*^9, 3.5609529577575006`*^9}, { 3.5609530081668854`*^9, 3.5609530111732287`*^9}, {3.56095334499853*^9, 3.560953363158815*^9}, {3.5609537938383837`*^9, 3.5609538133258977`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"-", "0.03918822487571427`"}], TraditionalForm]], "Output", CellChangeTimes->{ 3.560953011835338*^9, {3.560953357175073*^9, 3.5609533699857397`*^9}, { 3.560953797683914*^9, 3.5609538139709783`*^9}}] }, Open ]], Cell[TextData[{ "La solucion es un valor comprendido entre 2.5 y 2.55. Podemos calcularlo \ directamente con ", Cell[BoxData[ ButtonBox["FindRoot", BaseStyle->"Link", ButtonData->"paclet:ref/FindRoot"]]], " que se aplica cuando se desea resolver la ecuaci\[OAcute]n utilizando m\ \[EAcute]todos de aproximaci\[OAcute]n num\[EAcute]rica" }], "Text", CellChangeTimes->{{3.5609538502015686`*^9, 3.5609539735401535`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"1", " ", "+", " ", RowBox[{"Sin", "[", "x", "]"}], "-", SqrtBox["x"]}], "\[Equal]", "0"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "2.5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.5609534674260283`*^9, 3.5609535094473457`*^9}, { 3.560953543623702*^9, 3.5609535449508734`*^9}, {3.560953663408933*^9, 3.5609536903912754`*^9}, {3.5609537240785923`*^9, 3.5609537728947306`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"x", "\[Rule]", "2.5154557898966257`"}], "}"}], TraditionalForm]], "Output", CellChangeTimes->{{3.5609534925842113`*^9, 3.5609535116786895`*^9}, { 3.560953664584021*^9, 3.560953691333476*^9}, 3.5609537252446876`*^9, { 3.5609537641986265`*^9, 3.560953773871871*^9}}] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Ejercicios avanzados ", "Section", CellChangeTimes->{{3.561013799296636*^9, 3.561013802494676*^9}, { 3.5610187254144683`*^9, 3.5610187296552353`*^9}, {3.561018884818163*^9, 3.5610188974669275`*^9}, {3.5758124086500063`*^9, 3.5758124108432765`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ " Sea f definida por: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"Sen", "(", "x", ")"}], " ", "para", " ", "x"}], "\[LessEqual]", "c"}]}], TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", " ", RowBox[{ RowBox[{ RowBox[{"a", " ", "x"}], " ", "+", " ", RowBox[{"b", " ", "para", " ", "x"}]}], ">", "c"}]}], TraditionalForm]]], " siendo a, b, c constantes. Si b y c est\[AAcute]n dados, hallar todos los \ valores de a (si existe alguno) para los que f es continua en el punto c." }], "Subsubsection", CellChangeTimes->{{3.560055752443393*^9, 3.560055795499911*^9}, { 3.5600559141496706`*^9, 3.5600559225574265`*^9}, 3.5603176256627493`*^9, { 3.560317731993519*^9, 3.560317779371214*^9}, {3.575812432347077*^9, 3.575812434251335*^9}}, FontSize->18], Cell["Ejemplo: para c=1, a=2 y b=1", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "c_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "/;", RowBox[{"x", "\[LessEqual]", "c"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "c_"}], "]"}], ":=", " ", RowBox[{ RowBox[{ RowBox[{"a", " ", "x"}], " ", "+", " ", "b"}], "/;", RowBox[{"x", ">", "c"}]}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "==", " ", RowBox[{ RowBox[{"2", " ", "x"}], " ", "+", "1"}]}], ",", 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