Use Maple to determine the exact value of the limit
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Expression palette: Limit template
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Context Panel: Evaluate and Display Inline
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=
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Invoke the EpsilonDelta maplet
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1.
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Start the
EpsilonDelta
maplet and bring it to the state shown in Figure 1.2.4(a) by the following steps.
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In the top row of the interface, enter the function as , and enter , and in the appropriate windows.
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Set the plot ranges to , , , and .
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Set and .
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Click on the Plot button at the bottom of the Maplet window.
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Figure 1.2.4(a) EpsilonDelta maplet and
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Continue exploring the relationship between and .
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2.
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For , , and , find values of that satisfy the conditions of Definition 1.2.1.
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3.
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Test the formula . Do the choices made in step (2) agree with this formula?
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The astute observer will notice from Figure 1.2.4(a) that while the -band is symmetric around , the -band that is prescribed by Definition 1.2.1 is not uniformly filled by the corresponding horizontal blue band. If the horizontal blue band is to be symmetric with respect to , then the corresponding edges of the vertical blue band will not be symmetric with respect to . Hence, the beginnings of a computational strategy should start evolving. For an increasing function like , rather than try manipulating absolute values and inequalities, instead solve the equations and for and , then set . This process is formalized in Examples 1.2.5-8.