By Walter J Savitch
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Additional info for Abstract machines and grammars (Little, Brown computer systems series)
The function value at that point can be—but does not need to be—defined. This kind of discontinuity is nonremovable, because there is no way to redefine the function to connect the points. 4 y 3 2 Figure 3-3 1 -3 -2 -1 -1 x 1 2 3 There is a jump discontinuity at x = 1 on the graph in Figure 3-3. The limit as x approaches 1 from the left is 4, and the limit as x approaches 1 from the right is 2. Because the y values “jump” from 4 to 2, there is no way to redefine the y value at x = 1 to make the graph continuous.
And once that got too small, bring in the electron microscope! Even down to the atomic level, half of something is still something, even if it is too small for you to see or measure. The limit of the length of the remaining segment is zero inches, even though in theory it will never be completely gone. 0001 . . It is probably pretty clear that the numbers in the sequence are getting closer and closer to 3. Even though they will never actually reach 3, the numbers are approaching 3, and they can eventually get as close to 3 as you want; therefore, the limit is 3.
Removable Discontinuity An informal name for a removable discontinuity is a “hole” in a graph. You have already read about this kind of situation. When a factor cancels completely from the denominator of a rational function, there is a point missing from the graph at whatever value of x makes the canceled factor zero. ” The way to find the function value is to plug the value of x that makes the canceled factor zero into the reduced form of the rational function. It is also possible to have a piecewise-defined function that has a hole in it at a specific point of its domain.
Abstract machines and grammars (Little, Brown computer systems series) by Walter J Savitch