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Typically we present this precise notion of limit \ and then show how to use the definition to verify limits of (mostly) linear \ functions. Our students are then asked to do a few ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]-\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " limit proofs for linear functions. Most accomplish this task by \ recognizing that in all of our ``linear examples'' ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" is always given by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox["/|", Evaluatable->False, AspectRatioFixed->True], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["|, where f(x) = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "\[CapitalAE]x + b. The majority of them fail to understand the special \ relationship between the function's behavior near a point, the choice of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and a corresponding ``good'' ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Many might argue that the e-d definition of limit need not be dicussed in a \ first semester of calculus. Although the inuitive approach is certainly \ sufficient in the less rigorous calculus sequences, I feel we should expose \ our students to both the intuitive ideas of limit and the precise language of \ the e-d definition in any first semester calculus course which is part of the \ mathematics major. It is important in these early stages to stress that \ mathematics provides language which quantifies all of the terms and concepts \ introduced and used. This early exposure to the concreteness of mathematics \ helps strip away the ``trick up your sleeve'' mentality many students \ associate to mathematics and also lessens the impact of the brick wall of \ rigour students seem to hit in their upper level courses."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "However the presentation of the e-d definition in a first semester calculus \ course does not need to be entirely rigorous. Instead of requiring students \ to find a precise algebraic relationship between a function near a particular \ point, an arbitrary e and the ``best choice'' for d (except, perhaps, in the \ case of a linear function), it is enough for the students to grasp the idea \ that when a limit exists at a point, then for any positive value of e we can \ always find a value for d which satisfies the definition. There is plenty of \ time in later courses to determine an exact relationship between e and d. In \ first semester calculus we need only plant the seed for the precise e-d \ arguments. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A computer algebra system, such as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", can provide graphics and quick computations which allow students to \ discover that there is a relationship between an e and a suitable d in the \ definition of the limit. This note describes two ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" packages, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which can be used to supply graphical evidence that suitable d's can be \ found for any choice of a positive e whenever a function has a limit at a \ particular point.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Graphical approach to the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]-\[Delta]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], StyleBox[" definition", Evaluatable->False, AspectRatioFixed->True] }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["To ease the student into an ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]-\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " proof of the limit it is helpful to first let him/her explore the \ graphical relationship between different values for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and corresponding working values for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ". For example, suppose we propose to find the limit of the function f(x) \ = x", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " + 1 at x = 2. We could write out a rigorous proof that for any choice of \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = min{1,", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox["/5} shows that \nlim", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[RightArrow]", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" x", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " + 1 = 5. Most students will be asleep by the time we finish. Instead we \ could begin by asking the student to discover whether or not certain given \ values for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" are suitable for a prescribed ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ". To help the student accomplish this task, one could begin by asking the \ student to answer the following set of questions.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Consider the function f(x) = x^2 + 1."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["(a) What is the limit of f as x approaches 2?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) Let e=1. Does the choice of d=1 satisfy the definition of limit for \ f(x) = x^2 + 1, x->2? (I.e., is it true that for any x-value with the \ property |x-2|<1, then |f(x)-5|<1?) Explain."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(c) Again let e=1. Does the choice of d=1/10 satisfy the definition of \ limit for f(x) = x^2 + 1, x->2? (I.e., is it true that for any x-value with \ the property |x-2|<1/10, then |f(x)-5|<1?) Explain."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(d) Can you find (other) values for d when e=1 which satisfy the definition \ of limit for \nf(x) = x^2 + 1, x->2?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(e) Would the value d=min{1,1/5} for e=1 be suitable in the definition of \ limit for f(x) = x^2 + 1 as x->2?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The student could then gather more evidence by running through the same set \ of questions a second time with a different value for e and appropriately \ changed choices for d."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Without some kind of visual or graphical help, most students would find it \ very difficult to answer the above set of questions. This hardship can be \ alleviated with the appropriate graphical tools. As part of our growing ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "-based computer laboratory for calculus at Dickinson College, I have \ developed two such tools: the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" commands ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["LimitPlotDelta"], "Section", Evaluatable->False, PageBreakBelow->Automatic, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Let's return to our example: lim", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[RightArrow]", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" x", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + 1 = 5. Suppose the task is to determine whether or not ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox["=1 satisfies the definition of limit for the choice of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = 1. The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command can get the ball rolling. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" will graph the function in the specified ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ "-region in red, while the function outside of the region is drawn in \ black. The user can easily decide if the function's values in the region \ (2-", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[",2+", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[") fall in the light blue horizontal band of width 2", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " centered at the proposed limit. If the red portion of the graph falls \ outside the horizontal blue band, then the function has values farther away \ than ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox["=1 from the proposed limit somewhere in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ "-region around x=2; otherwise if the red portion of the graph of f is \ entirely enclosed in the blue horizontal band, then the function's values in \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox["-band about x=2 are guaranteed to be less than ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ "=1 away from the proposed limit of 5, and hence we have found a suitable ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox["=1. The usage statement for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is given below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?LimitPlotDelta"], "Input", AspectRatioFixed->True], Cell[TextData[ "LimitPlotDelta[f[x],x->a,L,epsilon,delta] will sketch the graph\n of \ y=f[x] near the point x=a, and will draw the horizontal\n lines \ y=L-epsilon and y=L+epsilon (in blue dashed lines), and\n will draw the \ vertical lines x=a-delta and x=a+delta (in red\n dashed lines)."], "Print",\ Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Returning to our example with f(x) = x", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->10, 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StyleBox[ "As we can see, the function's graph leaves the light blue horizontal \ region. (Recall that the light blue horizontal strip denotes the region (5-", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[", 5+", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ").) This indicates that we can find at least one x-value in the interval \ (2-", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[",2+", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ") which yields an f(x) value which is farther than 1 from 5. The number \ x=1.3 is such a value. 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by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["LimitPlotDelta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" conclusively shows that no suitable ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" can be found for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]", Evaluatable->False, AspectRatioFixed->True], StyleBox["=0.5, and hence the limit of f as x->0 can not be 0.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Conclusion"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Although gathering graphical evidence for the existence or nonexistence of \ a limit does not constitute an ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Epsilon]-\[Delta]", Evaluatable->False, 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\n On[Plot::plnr]\n\ ]"], "Input", InitializationCell->True, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10], Cell[TextData["End[];"], "Input", InitializationCell->True, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10], Cell[TextData["EndPackage[];"], "Input", InitializationCell->True, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10]}, Open]], Cell[CellGroupData[{Cell[TextData["About the author"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Shari Prevost is currently an Assistant Professor of Mathematics at \ Dickinson College. She received her Ph.D in mathematics from Rutgers \ University in the fall of 1989. Her research interests include affine Lie \ algebras and vertex operator theory."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\nShari Prevost\nDepartment of Mathematics and Computer Science\nDickinson \ College\nCarlisle, PA 17013\nprevost@dickinson.edu"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontSlant->"Italic", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{36, Automatic}, {Automatic, 31}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V:^/09R]g0000000OVaH097bCP0AP1Y06`0I@1^0642HZj` 0V:gT0000001nK500TO9>000000000000000009R[[0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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