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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 68231, 1898]*) (*NotebookOutlinePosition[ 69007, 1925]*) (* CellTagsIndexPosition[ 68963, 1921]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["p.504", "Subsection"], Cell[CellGroupData[{ Cell["21.", "Subsubsection"], Cell[TextData[{ "a) Show that ", Cell[BoxData[ \(ln \((x)\)\)]], " grows slower than ", Cell[BoxData[ \(x\^\(1/n\)\)]], "for any positive integer ", Cell[BoxData[ \(n\)]], "." }], "Text"], Cell[TextData[{ "Let ", Cell[BoxData[ \(a\ = \ 1/n\)]], ". Then ", Cell[BoxData[ \(lim\_\(x \[Rule] \[Infinity]\)x\^a\/\(ln \((x)\)\)\ = \ \(lim\_\(x \[Rule] \[Infinity]\)\ ax\^\(a - 1\)\/\(1/x\) = \(lim\_\(x \[Rule] \[Infinity]\)\ ax\^a\ = \ \[Infinity]\)\)\)]], "." }], "Text"], Cell[TextData[{ "b) Find an ", Cell[BoxData[ \(x\)]], " such that ", Cell[BoxData[ StyleBox[\(x\^\(1/1, 000, 000\)\ > \ ln \((x)\)\), "Text"]]], "." }], "Text"], Cell[TextData[{ "Try ", Cell[BoxData[ \(x\ = \ e\^\(10\^a\)\)]], ". Then ", Cell[BoxData[ \(x\^\(1/1, 000, 000\) = \ e\^\(10\^\(a - 6\)\)\)]], "and ", Cell[BoxData[ \(ln \((x)\)\ = \ 10\^a\)]], ". So we need to find a value of ", Cell[BoxData[ \(a\)]], " such that ", Cell[BoxData[ \(e\^\(10\^\(a - 6\)\) > \ 10\^a\)]], ". If ", Cell[BoxData[ \(a\ = \ 6\)]], ", ", Cell[BoxData[ \(e\^\(10\^\(a - 6\)\) = \ e\)]], ", which is not greater than ", Cell[BoxData[ \(10\^6\)]], ".\nIf ", Cell[BoxData[ \(a\ = \ 7\)]], ", then ", Cell[BoxData[ \(e\^\(10\^\(a - 6\)\) = \ e\^10\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[E\^10]\)], "Input"], Cell[BoxData[ \(22026.4657948067066`\)], "Output"] }, Open ]], Cell[TextData[{ "which is still not greater than ", Cell[BoxData[ \(10\^7\)]], ". But when ", Cell[BoxData[ \(a\ = \ 8\)]], ", we get ", Cell[BoxData[ \(e\^\(10\^\(a - 6\)\) = \ e\^100\)]], "," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[E\^100]\)], "Input"], Cell[BoxData[ \(2.68811714181612471`*^43\)], "Output"] }, Open ]], Cell[TextData[{ "which definitely is greater than ", Cell[BoxData[ \(10\^8\)]], ". Therefore, ", Cell[BoxData[ \(x\ = \ e\^\(10\^8\)\)]], "is a number such that ", Cell[BoxData[ \(x\^\(1/1, 000, 000\)\ > \ ln \(\((x)\) . \)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[E\^\(10\^8\)]\)], "Input"], Cell[BoxData[ \(1.5499767384052817158`7.9546*^43429448\)], "Output"] }, Open ]], Cell[TextData[{ "c) Find a value for ", Cell[BoxData[ \(x\)]], " where the graphs of ", Cell[BoxData[ \(x\^\(1/10\)\)]], "and ", Cell[BoxData[ \(ln \((x)\)\)]], " cross." }], "Text"], Cell[TextData[{ "The text is a litle imprecise here. There are two points with ", Cell[BoxData[ \(x\ > \ 1\)]], " where the graphs cross, one near 1 and the other a very large number. The \ text is asking you to find the latter. 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