数列极限的 ε-M 定义可视化
2014-08-28 06:03
本 GeoGebra 范例直观地演示了数列收敛的严格 ε-M（或 ε-N）定义。通过交互式界面，用户可以调节 ε（误差界限）和 M（项数阈值）等参数，动态观察数列各项（点）的分布情况。图形将数列的项区分为满足条件的“好点”与不满足条件的“坏点”，并高亮显示极限值的 ε 邻域。这有助于学生从几何角度深刻理解“对于任意给定的 ε > 0，总存在正整数 M，使得当 n > M 时，数列的所有项都落入极限的 ε 邻域内”这一微积分核心概念。
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<dimensions width="30" height="30" />
<caption val="+"/>
</element>
<expression label="vZoomLabel" exp=""Vertical zoom:""/>
<element type="text" label="vZoomLabel">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="500" y="15"/>
</element>
<expression label="hZoomLabel" exp=""Horizontal zoom:""/>
<element type="text" label="hZoomLabel">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
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<labelMode val="0"/>
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<absoluteScreenLocation x="627" y="16"/>
</element>
<expression label="fnAxisLabel" exp=""f_n""/>
<element type="text" label="fnAxisLabel">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="48" y="14"/>
</element>
<expression label="seqInputLabel" exp=""Enter the rule for the sequence here:""/>
<element type="text" label="seqInputLabel">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="107" y="17"/>
</element>
<expression label ="showEpsilon" exp="false"/>
<element type="boolean" label="showEpsilon">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="766" y="7"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="Show ε"/>
</element>
<expression label ="showM" exp="false"/>
<element type="boolean" label="showM">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="766" y="36"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="Show M"/>
</element>
<element type="button" label="hZoomInDummy">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="618" y="65"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[hScaleAnimStart,hScaleAnimEnd] SetValue[hScaleAnimEnd, hScaleAnimEnd/2] SetValue[ht,0] StartAnimation[ht] "/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="dummy +"/>
</element>
<expression label="def1" exp=""∀ε > 0""/>
<element type="text" label="def1">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
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<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="358" y="501"/>
</element>
<expression label="def2" exp=""∃M∈ℕ""/>
<element type="text" label="def2">
<show object="true" label="false"/>
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<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="412" y="501"/>
</element>
<expression label="def3" exp=""∀n >""/>
<element type="text" label="def3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="467" y="501"/>
</element>
<expression label="def4" exp=""M""/>
<element type="text" label="def4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="509" y="501"/>
</element>
<expression label="def7" exp=""ε""/>
<element type="text" label="def7">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="597" y="500"/>
</element>
<expression label="fnDisplay" exp=""f_n = " + (LaTeX[f]) + """ />
<element type="text" label="fnDisplay">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="310" y="10"/>
</element>
<expression label="def5" exp=""|f_n -""/>
<element type="text" label="def5">
<show object="true" label="false"/>
<objColor r="0" g="128" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="529" y="500"/>
</element>
<expression label="def6" exp=""<""/>
<element type="text" label="def6">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="581" y="501"/>
</element>
<element type="numeric" label="vScale">
<value val="5"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label ="g" exp="g(n) = 1 + (2 * (-1)^(n) / sqrt(n))"/>
<element type="function" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Textfield">
<input a0="g"/>
<output a0="gInput"/>
</command>
<element type="textfield" label="gInput">
<show object="true" label="true"/>
<condition showObject="showG"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="93" y="88"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<fixed val="true"/>
<auxiliary val="true"/>
<linkedGeo exp="g"/>
<length val="15"/>
<caption val="g_n ="/>
</element>
<expression label="gnDisplay" exp=""g_n = " + (LaTeX[g]) + """ />
<element type="text" label="gnDisplay">
<show object="true" label="false"/>
<condition showObject="showG"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="312" y="74"/>
</element>
<expression label ="showG" exp="false"/>
<element type="boolean" label="showG">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="524" y="75"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="Show second sequence"/>
</element>
<command name="Sequence">
<input a0="(n, g(n))" a1="n" a2="0" a3="nMax"/>
<output a0="GseqPoints"/>
</command>
<element type="list" label="GseqPoints">
<show object="true" label="false"/>
<condition showObject="((¬showEpsilon) ∨ (¬showM)) ∧ showG"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="4"/>
</element>
<command name="KeepIf">
<input a0="(x(P) > M) ∧ (abs(y(P) - L) ≥ epsilon)" a1="P" a2="GseqPoints"/>
<output a0="GbadPoints"/>
</command>
<element type="list" label="GbadPoints">
<show object="true" label="false"/>
<condition showObject="showEpsilon ∧ showM ∧ showG"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="4"/>
</element>
<command name="KeepIf">
<input a0="(x(P) > M) ∧ (abs(y(P) - L) < epsilon)" a1="P" a2="GseqPoints"/>
<output a0="GgoodPoints"/>
</command>
<element type="list" label="GgoodPoints">
<show object="true" label="false"/>
<condition showObject="showEpsilon ∧ showM ∧ showG"/>
<objColor r="51" g="255" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="4"/>
</element>
<command name="KeepIf">
<input a0="x(P) ≤ M" a1="P" a2="GseqPoints"/>
<output a0="GirrelevantPoints"/>
</command>
<element type="list" label="GirrelevantPoints">
<show object="true" label="false"/>
<condition showObject="showEpsilon ∧ showM ∧ showG"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="4"/>
</element>
<element type="point" label="Mpoint2">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1" y="2" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="Mpoint1" a1="Mpoint2"/>
<output a0="Mline"/>
</command>
<element type="line" label="Mline">
<show object="true" label="false" ev="4"/>
<condition showObject="showM"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="If[x(Mpoint1) < 0 || x(Mpoint2) < 0, SetCoords[Mpoint1, 0,1]] If[x(Mpoint1) < 0 || x(Mpoint2) < 0, SetCoords[Mpoint2, 0,2]]"/>
<coords x="-1" y="0" z="1"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<element type="point" label="Lpoint2">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2" y="0.9999" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="Lpoint1" a1="Lpoint2"/>
<output a0="Ldragline"/>
</command>
<element type="line" label="Ldragline">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="-0.9999"/>
<lineStyle thickness="2" type="20" typeHidden="1" opacity="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Ray">
<input a0="(0, L)" a1="(1, L)"/>
<output a0="LLine"/>
</command>
<element type="ray" label="LLine">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="-0.9999"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="epsilonPoint1">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1" y="1.496" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="epsilonPoint2">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2" y="1.496" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="epsilonPoint1" a1="epsilonPoint2"/>
<output a0="epsilonDragLine1"/>
</command>
<element type="line" label="epsilonDragLine1">
<show object="true" label="false" ev="4"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="SetCoords[epsilonPoint3,1,L+(L-y(epsilonPoint1))] SetCoords[epsilonPoint4,2,L+(L-y(epsilonPoint1))] SetValue[epsilon, abs(L-y(epsilonPoint1))]"/>
<coords x="0" y="1" z="-1.496"/>
<lineStyle thickness="2" type="20" typeHidden="1" opacity="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Ray">
<input a0="(0, L + epsilon)" a1="(1, L + epsilon)"/>
<output a0="epsilonLine1"/>
</command>
<element type="ray" label="epsilonLine1">
<show object="true" label="false" ev="4"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="-1.496"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="(0, L - epsilon)" a1="(1, L - epsilon)"/>
<output a0="epsilonLine2"/>
</command>
<element type="ray" label="epsilonLine2">
<show object="true" label="false" ev="4"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="-0.5038"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="epsilonPoint" exp="(0, L + epsilon)" type="point" />
<element type="point" label="epsilonPoint">
<show object="true" label="false"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="1.496" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="epsilonPoint3">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1" y="0.5038" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="epsilonPoint4">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2" y="0.5038" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="epsilonPoint3" a1="epsilonPoint4"/>
<output a0="epsilonDragLine2"/>
</command>
<element type="line" label="epsilonDragLine2">
<show object="false" label="false" ev="4"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="SetCoords[epsilonPoint1,1,L+(L-y(epsilonPoint3))] SetCoords[epsilonPoint2,2,L+(L-y(epsilonPoint3))] SetValue[epsilon, abs(L-y(epsilonPoint3))]"/>
<coords x="0" y="1" z="-0.5038"/>
<lineStyle thickness="2" type="20" typeHidden="1" opacity="0"/>
<eqnStyle style="implicit"/>
</element>
<expression label="epsilonLabel" exp="" ε = " + epsilon + """ />
<element type="text" label="epsilonLabel">
<show object="true" label="true"/>
<condition showObject="showEpsilon"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint exp="(viewLeft, L + epsilon / 2)"/>
</element>
<expression label="def8" exp=""L""/>
<element type="text" label="def8">
<show object="true" label="false"/>
<objColor r="51" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="565" y="500"/>
</element>
<command name="If">
<input a0="(y(Corner[1]) > 0) ∨ (y(Corner[4]) < 0)" a1="(0.9 * y(Corner[1])) + (0.1 * y(Corner[4]))" a2="0"/>
<output a0="Mycoord"/>
</command>
<element type="numeric" label="Mycoord">
<value val="0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="def9" exp=""|""/>
<element type="text" label="def9">
<show object="true" label="false"/>
<objColor r="0" g="128" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="575" y="500"/>
</element>
<command name="Vector">
<input a0="(M, Mycoord)" a1="(viewRight, Mycoord)"/>
<output a0="MRay"/>
</command>
<element type="vector" label="MRay">
<show object="true" label="false" ev="4"/>
<condition showObject="showM"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="19.4" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint x="1" y="0" z="1"/>
</element>
<expression label="MPoint" exp="(M, Mycoord)" type="point" />
<element type="point" label="MPoint">
<show object="true" label="true" ev="4"/>
<condition showObject="showM"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-16" y="35"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="M = %x"/>
</element>
</construction>
</geogebra>