Chapter 1: Limits
Section 1.2: Precise Definition of a Limit
Use the EpsilonDelta maplet to verify limx→2lnx=ln2.
Use Maple to determine the exact value of the limit
Expression palette: Limit template
Context Panel: Evaluate and Display Inline
limx→2lnx = ln⁡2
Invoke the EpsilonDelta maplet
maplet and bring it to the state shown in Figure 1.2.4(a) by the following steps.
In the top row of the interface, enter the function as lnx, and enter a=2, and L=ln2 in the appropriate windows.
Set the plot ranges to xmin=1, xmax=3, ymin=0, and ymax=2.
Set ϵ=0.80 and δ=0.30.
Click on the Plot button at the bottom of the Maplet window.
Figure 1.2.4(a) EpsilonDelta maplet and limx→2lnx=ln2
Continue exploring the relationship between ϵ and δ.
For ϵ=0.50, ϵ=0.25, and ϵ=0.10, find values of δ that satisfy the conditions of Definition 1.2.1.
Test the formula δϵ=2eϵ−1. Do the choices made in step (2) agree with this formula?
The astute observer will notice from Figure 1.2.4(a) that while the δ-band is symmetric around x=a, the ϵ-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 L, then the corresponding edges of the vertical blue band will not be symmetric with respect to x=a. Hence, the beginnings of a computational strategy should start evolving. For an increasing function like fx=lnx, rather than try manipulating absolute values and inequalities, instead solve the equations fa−δL=L−ϵ and fa+δR=L+ϵ for δL and δR, then set δ=minδL,δR. This process is formalized in Examples 1.2.5-8.
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