(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 462169, 8216] NotebookOptionsPosition[ 458382, 8096] NotebookOutlinePosition[ 458794, 8112] CellTagsIndexPosition[ 458751, 8109] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Heineho v\[EHacek]ta", "Title", CellChangeTimes->{{3.559277650857358*^9, 3.559277655152953*^9}}], Cell["\<\ Tom\[AAcute]\[SHacek] Kalvoda, KAM FIT \[CapitalCHacek]VUT, ZS 2013/2014\ \>", "Subtitle", CellChangeTimes->{{3.559277668384843*^9, 3.559277694335647*^9}, { 3.5594881338136*^9, 3.5594881384703074`*^9}, {3.5917643182577*^9, 3.591764321165942*^9}}], Cell["\<\ Heineho v\[EHacek]ta n\[AAcute]m n\[AAcute]m ukazuje jak mezi sebou souvis\ \[IAcute] pojmy \[OpenCurlyDoubleQuote]limita posloupnosti\ \[CloseCurlyDoubleQuote] a \[OpenCurlyDoubleQuote]limita funkce\ \[CloseCurlyDoubleQuote]. 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Jak je to s jej\[IAcute] limitou v bod\[EHacek] nula? To, \[ZHacek]e na \ obr\[AAcute]zku vid\[IAcute]me jen \[CHacek]ern\[YAcute] flek nic neznamen\ \[AAcute], co kdy\[ZHacek] na vhodn\[EHacek] zvolen\[EAcute] \ horizont\[AAcute]ln\[IAcute] \[SHacek]k\[AAcute]le ji\[ZHacek] funkce \ \[OpenCurlyDoubleQuote]zkonverguje\[CloseCurlyDoubleQuote] do nuly?" }], "Text", CellChangeTimes->{{3.559486580679789*^9, 3.5594866270781937`*^9}, { 3.559486946613933*^9, 3.5594870181357594`*^9}}], Cell[CellGroupData[{ Cell["Zoom", "Subsection", CellChangeTimes->{{3.559277882636044*^9, 3.559277883154727*^9}}], Cell["\<\ Pokusme se pod\[IAcute]vat na men\[SHacek]\[IAcute] okol\[IAcute] bodu nula, \ je snadn\[EAcute] p\[RHacek]edchoz\[IAcute] obr\[AAcute]zek rozpohybovat. 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obr\[AAcute]zku vid\[IAcute]me, \[ZHacek]e tato funkce sice po\[RHacek]\ \[AAcute]d velmi osciluje okolo nuly, kdy\[ZHacek] se nez\[AAcute]visle prom\ \[EHacek]nn\[AAcute] ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " bl\[IAcute]\[ZHacek]\[IAcute] k bodu ", Cell[BoxData[ FormBox["0", TraditionalForm]]], ". Ale d\[IAcute]ky tomu, \[ZHacek]e se n\[AAcute]sob\[IAcute] \ nez\[AAcute]visle prom\[EHacek]nnou ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ", tak je tato oscilace utolumena. Odhadujeme proto, \[ZHacek]e limita v \ nule existuje. To je jak ji\[ZHacek] v\[IAcute]me pravda, d\[IAcute]ky V\ \[EHacek]t\[EHacek] o limit\[EHacek] sev\[RHacek]en\[EAcute] funkce. Pod\ \[IAcute]vejme se na tento probl\[EAcute]m z pohledu Heineho v\[EHacek]ty." }], "Text", CellChangeTimes->{{3.559486580679789*^9, 3.5594866270781937`*^9}, { 3.559486946613933*^9, 3.5594870181357594`*^9}, {3.5594891477528057`*^9, 3.559489243419886*^9}}], Cell[CellGroupData[{ Cell["Dv\[EHacek] vhodn\[EAcute] posloupnosti", "Subsection", CellChangeTimes->{{3.55927799757537*^9, 3.559278004278493*^9}}], Cell["\<\ Zvolme op\[EHacek]t dv\[EHacek] posloupnosti, stejn\[EHacek] jako v minul\ \[EAcute]m odstavci.\ \>", "Text", CellChangeTimes->{{3.559487623272726*^9, 3.5594876341452837`*^9}, { 3.559489255177494*^9, 3.5594892845807962`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Definice", " ", "posloupnosti"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"x", "[", "n_", "]"}], "=", FractionBox["1", RowBox[{"2", "\[Pi]", " ", "n"}]]}], ";", RowBox[{ RowBox[{"z", "[", "n_", "]"}], "=", FractionBox["1", RowBox[{ RowBox[{"2", "\[Pi]", " ", "n"}], " ", "+", 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sou\[RHacek]adnic\[IAcute]ch ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "n"], ",", RowBox[{"g", "(", SubscriptBox["x", "n"], ")"}]}], ")"}], TraditionalForm]]], ", modr\[EAcute] punt\[IAcute]ky ozna\[CHacek]uj\[IAcute] body o \ sou\[RHacek]adnic\[IAcute]ch ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["z", "n"], ",", RowBox[{"g", "(", SubscriptBox["z", "n"], ")"}]}], ")"}], TraditionalForm]]], ". V demonstraci m\[URing]\[ZHacek]eme manipulovat indexem ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ". Vid\[IAcute]me, \[ZHacek]e kdy\[ZHacek] se ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " zv\[EHacek]t\[SHacek]uje, pak ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "-ov\[EAcute] sou\[RHacek]adnice bod\[URing] konverguj\[IAcute] k nule (", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "n"], "\[Rule]", "0"}], TraditionalForm]]], " a ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["z", "n"], "\[Rule]", "0"}], TraditionalForm]]], " kdy\[ZHacek] ", Cell[BoxData[ FormBox[ RowBox[{"n", "\[Rule]", "\[Infinity]"}], TraditionalForm]]], ") a posloupnosti ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-nov\[YAcute]ch sou\[RHacek]adnic, tedy funk\[CHacek]n\[IAcute]ch hodnot, \ ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"g", "(", SubscriptBox["x", "n"], ")"}], ")"}], TraditionalForm]]], "a ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"g", "(", SubscriptBox["z", "n"], ")"}], ")"}], TraditionalForm]]], " konverguj\[IAcute] ob\[EHacek] k ", Cell[BoxData[ FormBox["0", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.559488197972574*^9, 3.559488411403614*^9}, { 3.559551659346202*^9, 3.559551681378249*^9}, {3.559551712901599*^9, 3.559551713313919*^9}}], Cell[TextData[{ "Dodatek: Vid\[IAcute]me, \[ZHacek]e kdybychom zvolili posloupnost jinak, \ dostaneme jako limitu funk\[CHacek]n\[IAcute]ch hodnot v\[ZHacek]dy ", Cell[BoxData[ FormBox["0", TraditionalForm]]], ". Skute\[CHacek]n\[EHacek]. Bu\[DHacek] ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["u", "n"], ")"}], TraditionalForm]]], " libovoln\[AAcute] posloupnost nenulov\[YAcute]ch \[CHacek]len\[URing] \ konverguj\[IAcute]c\[IAcute] k bodu ", Cell[BoxData[ FormBox["0", TraditionalForm]]], ". Potom ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"0", "\[LessEqual]"}], "|", RowBox[{"g", "(", SubscriptBox["u", "n"], ")"}], "|"}], "=", RowBox[{ RowBox[{"|", RowBox[{ SubscriptBox["u", "n"], RowBox[{"sin", "(", RowBox[{"1", "/", SubscriptBox["u", "n"]}], ")"}]}], "|", RowBox[{"\[LessEqual]", RowBox[{"|", SubscriptBox["u", "n"], "|"}]}]}], "\[Rule]", "0"}]}], TraditionalForm]]], ", kdy\[ZHacek] ", Cell[BoxData[ FormBox[ RowBox[{"n", "\[Rule]", "\[Infinity]"}], TraditionalForm]]], ". Podle v\[EHacek]ty o limit\[EHacek] sev\[RHacek]en\[EAcute] posloupnosti \ proto je limita posloupnosti ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"g", "(", SubscriptBox["u", "n"], ")"}], ")"}], TraditionalForm]]], " rovna nule. Podle Heineho v\[EHacek]ty je tedy limitou funkce ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " v bod\[EHacek] ", Cell[BoxData[ FormBox["0", TraditionalForm]]], " \[CHacek]\[IAcute]slo ", Cell[BoxData[ FormBox["0", TraditionalForm]]], ". Na p\[RHacek]edn\[AAcute]\[SHacek]ce si tento fakt pozd\[EHacek]ji uk\ \[AAcute]\[ZHacek]eme i pomoc\[IAcute] v\[EHacek]ty o limit\[EHacek] sev\ \[RHacek]en\[EAcute] funkce." }], "Text", CellChangeTimes->{{3.559488415867168*^9, 3.559488463317309*^9}, { 3.559488663161909*^9, 3.55948870280722*^9}, {3.559551719369411*^9, 3.559551871439183*^9}, {3.559551923617339*^9, 3.559552010607082*^9}}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{681, 747}, WindowMargins->{{0, Automatic}, {2, Automatic}}, PrivateNotebookOptions->{"VersionedStylesheet"->{"Default.nb"[8.] -> False}}, FrontEndVersion->"9.0 for Linux x86 (64-bit) (January 25, 2013)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 103, 1, 92, "Title"], Cell[685, 25, 262, 5, 50, "Subtitle"], Cell[950, 32, 478, 8, 51, "Text"], Cell[1431, 42, 1790, 59, 93, "Text"], Cell[3224, 103, 891, 28, 53, "Text"], Cell[CellGroupData[{ Cell[4140, 135, 380, 12, 113, "Section"], Cell[4523, 149, 154, 2, 31, "Text", InitializationCell->True], Cell[4680, 153, 221, 6, 32, "Input"], Cell[4904, 161, 126, 1, 31, "Text"], Cell[CellGroupData[{ Cell[5055, 166, 1359, 29, 231, "Input"], Cell[6417, 197, 209552, 3447, 315, 179229, 2948, "CachedBoxData", "BoxData", \ "Output"] }, Open ]], Cell[215984, 3647, 786, 15, 91, "Text"], Cell[CellGroupData[{ Cell[216795, 3666, 92, 1, 44, "Subsection"], Cell[216890, 3669, 454, 8, 51, "Text"], Cell[CellGroupData[{ Cell[217369, 3681, 1655, 38, 275, "Input"], Cell[219027, 3721, 2058, 39, 406, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[221134, 3766, 126, 1, 44, "Subsection"], Cell[221263, 3769, 116, 1, 31, "Text"], Cell[221382, 3772, 652, 19, 143, "Input", InitializationCell->True], Cell[222037, 3793, 594, 16, 52, "Text"], Cell[CellGroupData[{ Cell[222656, 3813, 578, 16, 77, "Input"], Cell[223237, 3831, 122, 2, 32, "Output"], Cell[223362, 3835, 122, 2, 32, "Output"] }, Open ]], Cell[223499, 3840, 140, 1, 31, "Text"], Cell[CellGroupData[{ Cell[223664, 3845, 884, 24, 99, "Input"], Cell[224551, 3871, 92, 1, 32, "Output"], Cell[224646, 3874, 96, 1, 32, "Output"] }, Open ]], Cell[224757, 3878, 148, 3, 31, "Text"], Cell[CellGroupData[{ Cell[224930, 3885, 714, 19, 77, "Input"], Cell[225647, 3906, 192, 5, 32, "Output"], Cell[225842, 3913, 192, 5, 32, "Output"] }, Open ]], Cell[226049, 3921, 1070, 28, 92, "Text"], Cell[CellGroupData[{ Cell[227144, 3953, 891, 25, 77, "Input"], Cell[228038, 3980, 96, 1, 32, "Output"], Cell[228137, 3983, 96, 1, 32, "Output"] }, Open ]], Cell[228248, 3987, 296, 5, 51, "Text"], Cell[CellGroupData[{ Cell[228569, 3996, 435, 11, 55, "Input"], Cell[229007, 4009, 92, 1, 32, "Output"], Cell[229102, 4012, 92, 1, 32, "Output"] }, Open ]], Cell[229209, 4016, 302, 6, 31, "Text"], Cell[CellGroupData[{ Cell[229536, 4026, 2860, 68, 385, "Input"], Cell[232399, 4096, 2336, 50, 592, "Output"] }, Open ]], Cell[234750, 4149, 2138, 72, 135, "Text"], Cell[236891, 4223, 753, 16, 92, "Text"], Cell[CellGroupData[{ Cell[237669, 4243, 3341, 79, 551, "Input"], Cell[241013, 4324, 2604, 51, 618, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[243678, 4382, 455, 14, 113, "Section"], Cell[244136, 4398, 127, 1, 31, "Text"], Cell[244266, 4401, 325, 9, 32, "Input", InitializationCell->True], Cell[244594, 4412, 126, 1, 31, "Text"], Cell[CellGroupData[{ Cell[244745, 4417, 1410, 30, 231, "Input"], Cell[246158, 4449, 198490, 3265, 314, 177347, 2917, "CachedBoxData", \ "BoxData", "Output"] }, Open ]], Cell[444663, 7717, 974, 20, 111, "Text"], Cell[CellGroupData[{ Cell[445662, 7741, 126, 1, 44, "Subsection"], Cell[445791, 7744, 238, 5, 31, "Text"], Cell[446032, 7751, 625, 18, 143, "Input"], Cell[446660, 7771, 681, 17, 52, "Text"], Cell[447344, 7790, 148, 3, 31, "Text"], Cell[CellGroupData[{ Cell[447517, 7797, 760, 19, 77, "Input"], Cell[448280, 7818, 116, 2, 32, "Output"], Cell[448399, 7822, 116, 2, 32, "Output"] }, Open ]], Cell[448530, 7827, 358, 9, 31, "Text"], Cell[CellGroupData[{ Cell[448913, 7840, 2961, 70, 385, "Input"], Cell[451877, 7912, 2427, 50, 594, "Output"] }, Open ]], Cell[454319, 7965, 1980, 65, 115, "Text"], Cell[456302, 8032, 2040, 59, 160, "Text"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)