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Limits and Continuity

Limits and Continuity

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C. CONTINUITY AND DISCONTINUITY 1. One-sided limits We begin by expanding the notion of limit to include what are called one-sided limits, where z approaches a only from one side - the right or the left. The terminology and notation is:. right-hand limit lim f(x) (z comes from the right, x > a) left-hand limit lim f(z) (x comes from the left, z < a)X ML_ Since we use limits informally, a few examples will be enough to indicate the usefulness of this idea. 1/x 2 .1 1 Ex. 1 Ex.2 Ex. 3 Ex.4 Example 1. lim V/l- 2 = 0 =--1- =-*-1+ (As the picture shows, at the two endpoints of the domain, we only have a one-sided limit.) Example 2. Set f(z)= 1 z 0 Then lim f() = -1, lim f() =1.X>0. ,*---+ X-00+ Example 3. li -1 = oo, lim -1 = -oo -_O0+ X 2--0- 2
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