> > 範例︰使用極限運算子

1. 使用極限運算子，並以符號方式計算當期引數接近無限大時的運算式。
2. 繪製函數以促進其視覺化。使用水平標記來表示 e
 ns -30 Δn 0.01 ne 30 n ns ns Δn ne <legend xml:lang="zh_TW" /> <traces xml:lang="zh_TW"> <trace resultRef="41" xml:lang="zh_TW"> <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid" xml:lang="zh_TW">lines</traceStyle> </trace> </traces> <graph-size width="422.4" height="230.4" xml:lang="zh_TW" /> <axes xml:lang="zh_TW"> <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-30" end="30" xml:lang="zh_TW"> <axisLine position="origin" positionticmark="0" legendWidth="61.7533333333333" xml:lang="zh_TW" /> <axisGrid xml:lang="zh_TW"> <gridFrequency xml:lang="zh_TW">7</gridFrequency> <gridLabels display="true" xml:lang="zh_TW" /> <gridLines xml:lang="zh_TW" /> <tickMarks display="true" xml:lang="zh_TW" /> </axisGrid> <axisLabel xml:lang="zh_TW" /> <markers xml:lang="zh_TW"> <marker resultRef="42" display="true" color="#FF000000" value="-1" xml:lang="zh_TW" line-style="Dash"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">-1</real> </marker> </markers> <plotEquations xml:lang="zh_TW"> <plotEquation xml:lang="zh_TW"> <math xml:lang="zh_TW"> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">n</id> </math> <math xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW"> <startValue xml:lang="zh_TW" resultRef="43"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">-30</real> </startValue> <secondTickValue resultRef="44" xml:lang="zh_TW"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">-20</real> </secondTickValue> <endValue xml:lang="zh_TW" resultRef="45"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">30</real> </endValue> </xyDomain> </xAxis> <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="7" xml:lang="zh_TW"> <axisLine position="origin" positionticmark="3" legendWidth="110.17" xml:lang="zh_TW" /> <axisGrid xml:lang="zh_TW"> <gridFrequency xml:lang="zh_TW">8</gridFrequency> <gridLabels display="true" xml:lang="zh_TW" /> <gridLines xml:lang="zh_TW" /> <tickMarks display="true" xml:lang="zh_TW" /> </axisGrid> <axisLabel xml:lang="zh_TW" /> <markers xml:lang="zh_TW"> <marker resultRef="46" display="true" color="#FF068149" value="2.7182818284590451" xml:lang="zh_TW" line-style="Dash"> <id xml:space="preserve" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">e</id> </marker> <marker resultRef="47" display="true" color="#FF662D91" value="1" xml:lang="zh_TW" line-style="Dash"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">1</real> </marker> </markers> <plotEquations xml:lang="zh_TW"> <plotEquation xml:lang="zh_TW"> <math xml:lang="zh_TW"> <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <pow xml:lang="zh_TW" /> <parens xml:lang="zh_TW"> <apply xml:lang="zh_TW"> <plus xml:lang="zh_TW" /> <real xml:lang="zh_TW">1</real> <apply xml:lang="zh_TW"> <div xml:lang="zh_TW" /> <real xml:lang="zh_TW">1</real> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">n</id> </apply> </apply> </parens> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">n</id> </apply> </math> <math xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW"> <startValue xml:lang="zh_TW" resultRef="48"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">0</real> </startValue> <secondTickValue resultRef="49" xml:lang="zh_TW"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">1</real> </secondTickValue> <endValue xml:lang="zh_TW" resultRef="50"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">7</real> </endValue> </xyDomain> </yAxis> </axes> </xyPlot> </plot> </region>
(x, y) 象限中，我們觀察下列項目︰

 n 1
1 n
n
e
n 1
1 n
n
0
1
(-x, y) 象限中，我們觀察下列項目︰

 n 1
1 n
n
e
n 1
1 n
n
-1
 使用第二個方程式中的左側極限方向，表示將從曲線的左側接近 -1。如果未指定，則因為未針對 -1 < n < 0 定義函數，因此計算將會傳回「未定義」： n 1
1 n
n
-1
n
1 n
1
n
-1

1. 繪製 cot 函數。
 xs -3.0 Δx 0.1 xe 3.0 x xs xs Δx xe <legend xml:lang="zh_TW" /> <traces xml:lang="zh_TW"> <trace resultRef="60" xml:lang="zh_TW"> <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid" xml:lang="zh_TW">lines</traceStyle> </trace> </traces> <graph-size width="440.861538461538" height="230.4" xml:lang="zh_TW" /> <axes xml:lang="zh_TW"> <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-5" end="5" xml:lang="zh_TW"> <axisLine position="origin" positionticmark="5" legendWidth="60.93" xml:lang="zh_TW" /> <axisGrid xml:lang="zh_TW"> <gridFrequency xml:lang="zh_TW">5</gridFrequency> <gridLabels display="true" xml:lang="zh_TW" /> <gridLines xml:lang="zh_TW" /> <tickMarks display="true" xml:lang="zh_TW" /> </axisGrid> <axisLabel xml:lang="zh_TW" /> <markers xml:lang="zh_TW"> <marker resultRef="61" display="true" color="#FF000000" value="3.1415926535897931" xml:lang="zh_TW" line-style="Dash"> <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">π</id> </marker> <marker resultRef="62" display="true" color="#FF000000" value="-3.1415926535897931" xml:lang="zh_TW" line-style="Dash"> <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <neg xml:lang="zh_TW" /> <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">π</id> </apply> </marker> </markers> <plotEquations xml:lang="zh_TW"> <plotEquation xml:lang="zh_TW"> <math xml:lang="zh_TW"> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">x</id> </math> <math xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW"> <startValue xml:lang="zh_TW" resultRef="63"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">-5</real> </startValue> <secondTickValue resultRef="64" xml:lang="zh_TW"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">-2.5</real> </secondTickValue> <endValue xml:lang="zh_TW" resultRef="65"> <real xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">5</real> </endValue> </xyDomain> </xAxis> <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10" xml:lang="zh_TW"> <axisLine position="origin" positionticmark="2" legendWidth="96.75" xml:lang="zh_TW" /> <axisGrid xml:lang="zh_TW"> <gridFrequency xml:lang="zh_TW">11</gridFrequency> <gridLabels display="true" xml:lang="zh_TW" /> <gridLines xml:lang="zh_TW" /> <tickMarks display="true" xml:lang="zh_TW" /> </axisGrid> <axisLabel xml:lang="zh_TW" /> <markers xml:lang="zh_TW" /> <plotEquations xml:lang="zh_TW"> <plotEquation xml:lang="zh_TW"> <math xml:lang="zh_TW"> <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <id labels="FUNCTION" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">cot</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">x</id> </apply> </math> <math xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW"> <startValue xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </startValue> <endValue xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" /> </endValue> </xyDomain> </yAxis> </axes> </xyPlot> </plot> </region>
(x, y) 象限中，我們觀察下列項目︰

 x cot x 0 x cot x 0 x cot x π x cot x π

2. 指定「極限方向」，然後以符號方式重新計算圍繞 0πcot 函數。
 x cot x 0 x cot x 0 x cot x π x cot x π

 有時它有助於繪製函數，以實現其視覺化，以及檢驗符號計算結果的有效性。