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Notes 2C (Continuity)

Notes 2C (Continuity) - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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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.) z 0 Example 2. Set f(z)= 1 X 0. Then > ,*---+ lim f() = -1, X-00+ lim f() = 1. 1 1 Example 3. li - = oo, lim - = -oo -_O0+ X 2--0- 2 1 1 Example 4. li n- =oo, lim - oo .-- 4 0+ 2 2 2-o- z The relationship between the one-sided limits and the usual (two-sided) limit. is given by (1) . IC-=+& lim f(x) = L lim f(x) = L and lim f() = L 2-40+ In words, the (two-sided) limit exists if and only if both one-sided limits exist and are equal. This shows for example that in Examples 2 and 3 above, lim f(z) does not exist. Students often say carelessly that lim 1/x = oo, but this is not sloppy, it is simply 2--+0 wrong, as the picture for Example 3 shows. By contrast, lirn 1/z = oo is correct and acceptable terminology.
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