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embizze eNotes educator| Certified Educator

We are given a graph of a piece-wise defined function:

(a) f(0)=1 The function is defined at zero, and the value is 1. Note that there is a solid dot at (0,1), while there is an open dot at (0,4). The open dot indicates that the function is not defined there.

(b) `lim_(x->0)f(x) ` does not exist. The limit as you approach zero from the left is 4 while the limit from the right is 1. Since the left hand and right hand limits do not agree, the limit does not exist. (There is a jump discontinuity at x=0.)

(c) f(2) does not exist. There is an open dot at (2,3) and no closed dot for x=2. (There is a removable point discontinuity at x=2.)

(d) ` lim_(x->2)f(x)=3 ` The limit at x=2 exists and equals 3. Using the graph we see that f(x) gets arbitrarily close to 3 as x gets close to 2. (This is a naive definition of limit -- we would need the function definition to formally show the existence of the limit.)

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