双变量函数极限不存在的双路径法演示
2015-01-05 06:27
本范例演示了如何使用双路径法证明双变量函数的极限不存在。通过构造两条不同的路径趋近于目标点,展示函数在这些路径上的极限值不同,从而直观地说明该二元函数在指定点的极限不存在。适用于微积分课程中多元函数极限概念的教学与理解。
所属专题
微积分
GeoGebra 画板及绘图内容加载中,请稍等……
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.42.0" id="11f63d28-0dbd-4fd0-b337-b222e0acb17e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.185" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="855,257,700,601" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3,3" size="300" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="148" window="58,631,600,251" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3" size="200" window="859,28,250,598" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 22 , 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="true" inframe="false" stylebar="false" location="1" size="632" window="100,100,600,400" />
</views>
<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="148" height="471"/>
<coordSystem xZero="564.9999999999998" yZero="359.00000000000017" scale="50.00000000000004" yscale="50.00000000000003"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<spreadsheetView>
</spreadsheetView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.3167350779470194" yZero="0.03163186959781081" zZero="0.35716152203453255" scale="117.70873706336619" xAngle="26.0" zAngle="1106.8470842882607"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="0"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<command name="PointIn">
<input a0="z = 0"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Point">
<input a0="zAxis"/>
<output a0="Pz"/>
</command>
<element type="point3d" label="Pz">
<show object="true" label="false"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="-1.2289245350641567" w="1.0"/>
<pointSize val="3"/>
</element>
<element type="vector3d" label="w">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Point">
<input a0="Line[A, A + w]"/>
<output a0="D"/>
</command>
<element type="point3d" label="D">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="3.515234013969843" w="1.0"/>
<pointSize val="3"/>
</element>
<element type="numeric" label="k">
<value val="-0.3"/>
<show object="false" label="true"/>
<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 ="f" exp="f(x, y) = ((x * y^(2))) / (x^(2) + y^(4))"/>
<element type="functionNVar" label="f">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<levelOfDetail val="9"/>
</element>
<element type="vector3d" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="1.0" y="0.0" z="0.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<element type="vector3d" label="v">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="1.0" z="0.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Point">
<input a0="Line[A, A + u]"/>
<output a0="B1"/>
</command>
<element type="point3d" label="B1">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="2.227998976729099" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Point">
<input a0="Line[A, A + v]"/>
<output a0="C1"/>
</command>
<element type="point3d" label="C1">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="-1.882787953010927" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<expression label="wx" exp="abs(x(B1 - A))" />
<element type="numeric" label="wx">
<value val="2.227998976729099"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="wy" exp="abs(y(C1 - A))" />
<element type="numeric" label="wy">
<value val="1.882787953010927"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Point">
<input a0="Line[A, A + u]"/>
<output a0="B2"/>
</command>
<element type="point3d" label="B2">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="-1.620773163945259" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Min">
<input a0="x(B1)" a1="x(B2)"/>
<output a0="a1"/>
</command>
<element type="numeric" label="a1">
<value val="-1.620773163945259"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="x(B1)" a1="x(B2)"/>
<output a0="a2"/>
</command>
<element type="numeric" label="a2">
<value val="2.227998976729099"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Point">
<input a0="Line[A, A + v]"/>
<output a0="C2"/>
</command>
<element type="point3d" label="C2">
<show object="false" label="true"/>
<condition showObject="showControls"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="1.9220274942725342" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Min">
<input a0="y(C1)" a1="y(C2)"/>
<output a0="b1"/>
</command>
<element type="numeric" label="b1">
<value val="-1.882787953010927"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="y(C1)" a1="y(C2)"/>
<output a0="b2"/>
</command>
<element type="numeric" label="b2">
<value val="1.9220274942725342"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Function">
<input a0="f(s, t)" a1="s" a2="a1" a3="a2" a4="t" a5="b1" a6="b2"/>
<output a0="f2"/>
</command>
<element type="functionNVar" label="f2">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<levelOfDetail val="9"/>
</element>
<command name="Polygon">
<input a0="(a1, b1)" a1="(a1, b2)" a2="(a2, b2)" a3="(a2, b1)"/>
<output a0="poly1" a1="seg11" a2="seg12" a3="seg13" a4="seg14"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="seg11">
<show object="false" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-3.8048154472834614" y="0.0" z="-6.166742770721411"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="seg12">
<show object="false" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="3.848772140674358" z="-7.397445873566274"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="seg13">
<show object="false" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="3.8048154472834614" y="0.0" z="-8.47712492319062"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="seg14">
<show object="false" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="-3.848772140674358" z="-7.246421820345758"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="PointIn">
<input a0="poly1"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.9951658875515856" y="0.4136493142665949" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="Q" exp="P + (0, 0, f(x(P), y(P)))" type="point" />
<element type="point3d" label="Q">
<show object="false" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.9951658875515856" y="0.4136493142665949" z="0.16700001251093197" w="1.0"/>
<pointSize val="2"/>
</element>
<command name="CurveCartesian">
<input a0="t^(2)" a1="t" a2="f(t^(2), t)" a3="t" a4="y(C1)" a5="0"/>
<output a0="curve1"/>
</command>
<element type="curvecartesian3d" label="curve1">
<show object="true" label="false"/>
<objColor r="128" g="0" b="128" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label="z0" exp="z(Pz)" />
<element type="numeric" label="z0">
<value val="-1.2289245350641567"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="planeXY" exp="z = z0" type="plane" />
<element type="plane3d" label="planeXY">
<show object="true" label="false"/>
<objColor r="160" g="160" b="160" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="1.2289245350641567"/>
<fading val="0.0"/>
</element>
<element type="boolean" label="showControls">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="5" y="5"/>
<labelMode val="0"/>
</element>
<command name="Vector">
<input a0="(x(B2), y(B2), z0)" a1="(x(B1), y(B1), z0)"/>
<output a0="vAxisX"/>
</command>
<element type="vector3d" label="vAxisX">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="3.848772140674358" y="0.0" z="0.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint x="-1.620773163945259" y="0.0" z="-1.2289245350641567" w="1.0"/>
</element>
<command name="Vector">
<input a0="(x(C1), y(C1), z0)" a1="(x(C2), y(C2), z0)"/>
<output a0="vAxisY"/>
</command>
<element type="vector3d" label="vAxisY">
<show object="true" label="false"/>
<objColor r="0" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="3.8048154472834614" z="0.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint x="0.0" y="-1.882787953010927" z="-1.2289245350641567" w="1.0"/>
</element>
<command name="CurveCartesian">
<input a0="t^(2)" a1="t" a2="z0" a3="t" a4="y(C1)" a5="0"/>
<output a0="path1"/>
</command>
<element type="curvecartesian3d" label="path1">
<show object="true" label="false"/>
<objColor r="128" g="0" b="128" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="CurveCartesian">
<input a0="t" a1="t" a2="f(t, t)" a3="t" a4="0" a5="y(C2)"/>
<output a0="curve2"/>
</command>
<element type="curvecartesian3d" label="curve2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="CurveCartesian">
<input a0="t" a1="t" a2="z0" a3="t" a4="0" a5="y(C2)"/>
<output a0="path2"/>
</command>
<element type="curvecartesian3d" label="path2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label="text1" exp=""f(x,y)=" + (LaTeX[f]) + """ />
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="11" y="53"/>
</element>
<command name="Point">
<input a0="path1"/>
<output a0="PP1"/>
</command>
<element type="point3d" label="PP1">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.9999999529593336" y="-0.9999999764796665" z="-1.2289245350641567" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Point">
<input a0="path2"/>
<output a0="PP2"/>
</command>
<element type="point3d" label="PP2">
<show object="true" label="false"/>
<objColor r="255" g="102" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="1.0" y="1.0" z="-1.2289245350641567" w="1.0"/>
<pointSize val="4"/>
</element>
<expression label="PC1" exp="(x(PP1), y(PP1), f(x(PP1), y(PP1)))" type="point" />
<element type="point3d" label="PC1">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.9999999529593336" y="-0.9999999764796665" z="0.5" w="1.0"/>
<pointSize val="4"/>
</element>
<expression label="PC2" exp="(x(PP2), y(PP2), f(x(PP2), y(PP2)))" type="point" />
<element type="point3d" label="PC2">
<show object="true" label="false"/>
<objColor r="255" g="102" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.0" y="1.0" z="0.5" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Segment">
<input a0="PC1" a1="PP1"/>
<output a0="a"/>
</command>
<element type="segment3d" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="10" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="PC2" a1="PP2"/>
<output a0="b"/>
</command>
<element type="segment3d" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="10" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>