Chapter 1: Limits
Section 1.2: Precise Definition of a Limit
Use the EpsilonDelta maplet to verify limx→3(3⁢x−4)=5.
maplet and bring it to the state shown in Figure 1.2.1(a) by the following steps.
In the top row of the interface, enter the function as 3*x - 4, and enter a=3, and L=5 in the appropriate windows.
Set the plot ranges to xmin=0, xmax=5, ymin=0, and ymax=10.
Set ϵ=0.50 and δ=0.50 (use the slider to come close to these values).
Click on the Plot button at the bottom of the Maplet window.
Figure 1.2.1(a) EpsilonDelta maplet and limx→3(3⁢x−4)=5
Note that the blue horizontal strip is not contained within the red lines.
The value δ=0.50 is too large to satisfy the conditions in Definition 1.2.1.
Continue exploring the relationship between ϵ and δ.
Change the value of δ to 0.05.
Now the blue horizontal strip falls entirely within the red lines. This value of δ does satisfy the conditions in the definition of the limit.
But, this is not enough! The definition states that it must be possible to find such a δ for every ϵ>0. At present, this property has been shown true only for ϵ=0.5.
Find (exactly or approximately) the largest value of δ for which the conditions of the definition of the limit are satisfied for ϵ = 0.50.
Repeat step (3) for ϵ = 0.25. (Try δ = 0.1 and δ = 0.05 to get started.)
Is it possible to conjecture a general rule for selecting δ for any given ϵ?
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