Chapter 1: Limits
Section 1.2: Precise Definition of a Limit
Use the EpsilonDelta maplet to verify limx→3x2−3⁢x+3=3.
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
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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