Chapter 1: Limits
Section 1.1: Naive Limits
Example 1.1.6
Obtain limx→1gx, if g is the piecewise function given by gx=2−x2x<1x−1− x+2x≥1.
Solution
Control-drag gx=… Context Panel: Assign Function
(Click here to see a detailed explanation of how to enter a piecewise-defined function in Maple.)
gx=2−x2x<1x−1− x+2x≥1→assign as functiong
Figure 1.1.6(a), while not essential, is extremely revealing about the behavior of gx near x=1. The jump in the function shows that the two one-sided limits will not be equal, and that the limit itself will not exist.
Code for drawing Figure 1.1.6(a) is hidden behind the cell containing it. It can also be drawn interactively via the Plot Builder. (The relevant options are: in Basic Options set the range for x, and in Global Options set the view for axis[2].)
Figure 1.1.6(a) Graph of piecewise-defined gx
The left-hand rule in g can be obtained by simplifying g under the assumption that x<1.
Context Panel: Evaluate at a Point≻x=1
The right-hand rule in g can be obtained by simplifying x≥1.
simplifygx assuming x<1
−x2+2
→evaluate at point
1
simplifygx assuming x≥1
x−1−x+2
The limit from the left is the value of the left-hand rule taken at x=1.
The limit from the right is the value of the right-hand rule taken at x=1.
Expression palette: Limit template Obtain the limit from the left.
Expression palette: Limit template Obtain the limit from the right.
limx→1−gx = 1
limx→1+gx = 1
The limit of x→1
Expression palette: Limit template Obtain the limit of g.
limx→1gx = 1
For this function, limx→1gx exists because the two one-sided limits agree. The value of the limit is the common value of the two one-sided limits.
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