Operators > Symbolic Operators > Example: Using the Limit Operator
Example: Using the Limit Operator
Using Infinity as a Limiting Value
1. Use the limit operator and symbolically evaluate an expression as its argument approaches infinity.
2. Plot the function to facilitate its visualization. Use a horizontal marker to represent e.
 ns -30 Δn 0.01 ne 30 n ns ns Δn ne <legend /> <traces> <trace resultRef="41"> <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle> </trace> </traces> <graph-size width="422.4" height="230.4" /> <axes> <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-30" end="30"> <axisLine position="origin" positionticmark="0" legendWidth="61.7533333333333" /> <axisGrid> <gridFrequency>7</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <markers> <marker resultRef="42" display="true" color="#FF000000" line-style="Dash" value="-1"> <real xmlns="http://schemas.mathsoft.com/math50">-1</real> </marker> </markers> <plotEquations> <plotEquation> <math> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">n</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true"> <startValue resultRef="43"> <real xmlns="http://schemas.mathsoft.com/math50">-30</real> </startValue> <secondTickValue resultRef="44"> <real xmlns="http://schemas.mathsoft.com/math50">-20</real> </secondTickValue> <endValue resultRef="45"> <real xmlns="http://schemas.mathsoft.com/math50">30</real> </endValue> </xyDomain> </xAxis> <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="7"> <axisLine position="origin" positionticmark="3" legendWidth="110.17" /> <axisGrid> <gridFrequency>8</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <markers> <marker resultRef="46" display="true" color="#FF068149" line-style="Dash" value="2.7182818284590451"> <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">e</id> </marker> <marker resultRef="47" display="true" color="#FF662D91" line-style="Dash" value="1"> <real xmlns="http://schemas.mathsoft.com/math50">1</real> </marker> </markers> <plotEquations> <plotEquation> <math> <apply xmlns="http://schemas.mathsoft.com/math50"> <pow /> <parens> <apply> <plus /> <real>1</real> <apply> <div /> <real>1</real> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n</id> </apply> </apply> </parens> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n</id> </apply> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true"> <startValue resultRef="48"> <real xmlns="http://schemas.mathsoft.com/math50">0</real> </startValue> <secondTickValue resultRef="49"> <real xmlns="http://schemas.mathsoft.com/math50">1</real> </secondTickValue> <endValue resultRef="50"> <real xmlns="http://schemas.mathsoft.com/math50">7</real> </endValue> </xyDomain> </yAxis> </axes> </xyPlot> </plot> </region>
In the (x, y) quadrant we observe the following:
As n approaches positive infinity, the function approaches y=e.
As n approaches 0, the function approaches y=1.
Mathematically, this is represented by the following symbolic evaluations:
 n 1
1 n
n
e
n 1
1 n
n
0
1
In the (-x, y) quadrant we observe the following:
As n approaches negative infinity, the function approaches y=e.
As n approaches -1, the function approaches y=infinity.
Mathematically, this is represented by the following symbolic evaluations:
 n 1
1 n
n
e
n 1
1 n
n
-1
 The use of Left-hand Limit-side in the second equation means that the -1 is to be approached from the left-side of the curve. If this is not specified, then the evaluation returns "undefined" because the function is not defined for -1 < n < 0: n 1
1 n
n
-1
n
1 n
1
n
-1
Using the Limit Side
1. Plot the cot function.
 xs -3.0 Δx 0.1 xe 3.0 x xs xs Δx xe <legend /> <traces> <trace resultRef="60"> <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle> </trace> </traces> <graph-size width="440.861538461538" height="230.4" /> <axes> <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-5" end="5"> <axisLine position="origin" positionticmark="5" legendWidth="60.93" /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <markers> <marker resultRef="61" display="true" color="#FF000000" line-style="Dash" value="3.1415926535897931"> <id xml:space="preserve" labels="CONSTANT" xmlns="http://schemas.mathsoft.com/math50">π</id> </marker> <marker resultRef="62" display="true" color="#FF000000" line-style="Dash" value="-3.1415926535897931"> <apply xmlns="http://schemas.mathsoft.com/math50"> <neg /> <id xml:space="preserve" labels="CONSTANT">π</id> </apply> </marker> </markers> <plotEquations> <plotEquation> <math> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">x</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true"> <startValue resultRef="63"> <real xmlns="http://schemas.mathsoft.com/math50">-5</real> </startValue> <secondTickValue resultRef="64"> <real xmlns="http://schemas.mathsoft.com/math50">-2.5</real> </secondTickValue> <endValue resultRef="65"> <real xmlns="http://schemas.mathsoft.com/math50">5</real> </endValue> </xyDomain> </xAxis> <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10"> <axisLine position="origin" positionticmark="2" legendWidth="96.75" /> <axisGrid> <gridFrequency>11</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <markers /> <plotEquations> <plotEquation> <math> <apply xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">cot</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id> </apply> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <xyDomain scale-type="linear" auto-scale="true"> <startValue> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </startValue> <endValue> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </endValue> </xyDomain> </yAxis> </axes> </xyPlot> </plot> </region>
In the (x, y) quadrant we observe the following:
As x approaches 0, the function approaches y=infinity.
As x approaches π, the function approaches y=-infinity.
Mathematically, this is represented by the following symbolic evaluations:
 x cot x 0 x cot x 0 x cot x π x cot x π
Since the function is symmetric around x=+/- n*π/2, the symbolic evaluation returns "undefined" because the function around x=0 (and any multiple number of π) can be either infinity or -infinity, depending on the side from which x approaches 0.
This is a good case for specifying the "Limit Side".
2. Specify the "Limit Side" and symbolically reevaluate the cot function around 0 and π.
 x cot x 0 x cot x 0 x cot x π x cot x π
The returned results agree with the plot.
 Sometimes it helps to plot a function in order to visualize it and to double check the validity of symbolic evaluation results.