Chapter 1: Limits
Section 1.2: Precise Definition of a Limit
Example 1.2.3
Use the EpsilonDelta maplet to verify limx→3x2−3x+3=3.
Solution
Use Maple to determine the exact value of the limit
Expression palette: Limit template
Context Panel: Evaluate and Display Inline
limx→3x2−3 x+3 = 3
Invoke the EpsilonDelta maplet
Start the EpsilonDelta maplet and bring it to the state shown in Figure 1.2.3(a) by the following steps.
In the top row of the interface, enter the function as x^2-3*x+3, and enter a=3, and L=3 in the appropriate windows.
Set the plot ranges to xmin=1, xmax=4, ymin=1, and ymax=7.
Set ϵ=0.80 and δ=0.20.
Click on the Plot button at the bottom of the Maplet window.
Figure 1.2.3(a) EpsilonDelta maplet and limx→3x2−3 x+3=3
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.
Evaluate x2−3 x+3 at x=3.
In Example 1.2.9 a general formula for δ=δϵ is found. Warning: this is a challenging affair.
The astute observer will note that the horizontal blue band in Figure 1.2.3(a) is not symmetrically placed within the (red) lines y=3 ±ϵ. This is because the edges of the vertical blue band are symmetrically placed at x=3 ±δ. Because the graph of x2−3 x+3 is curved, the values f3 ± δ are not uniformly spaced above and below the line y=L=3. See Example 1.2.8 for a full determination of δ as a function of ϵ.
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