SECTION 14.3 One-sided Limits; Continuity 955 Finding One-sided Limits of a Function For the function f x x x x x x 2 1 if 2 1 if 2 2 if 2 ( ) = − < = − > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ find: (a) f x lim x 2 ( ) → − (b) f x lim x 2 ( ) → + (c) f x lim x 2 ( ) → Solution EXAMPLE 1 Figure 9 shows the graph of f. (a) To find f x lim , x 2 ( ) → − look at the values of f when x is close to 2 but less than 2. Since f x x2 1 ( ) = − for such numbers, we conclude that f x x lim lim 2 1 3 x x 2 2 ( ) ( ) = − = → → − − (b) To find f x lim , x 2 ( ) → + look at the values of f when x is close to 2 but greater than 2. Since f x x 2 ( ) = − for such numbers, we conclude that f x x lim lim 2 0 x x 2 2 ( ) ( ) = − = → → + + (c) Since the left and right limits are unequal, f x lim x 2 ( ) → does not exist. Figure 9 x y 2 22 22 2 (2, 1) 4 Now Work PROBLEMS 21 AND 35 2 Determine Whether a Function Is Continuous at a Number We have observed that f c , ( ) the value of the function f at c, plays no role in determining the one-sided limits of f at c. What is the role of the value of a function at c and its one-sided limits at c? Let’s look at some of the possibilities. See Figure 10. Figure 10 (b) x y lim f (x) 5 lim f (x), so lim f (x) exists; lim f(x) 5 f(c) f(c) y 5 f (x) c2 c x (a) c1 x x c x c x y lim f(x) ? lim f(x), so lim f(x) does not exist; f(c) is defined f (c) y 5 f (x) c2 c x (d) c1 x x c x y lim f (x) 5 lim f (x), so lim f (x) exists; lim f(x) ? f(c) f(c) y 5 f (x) c2 c x c1 x x c x c x y lim f(x) ? lim f(x), so lim f(x) does not exist; f(c) is not defined y 5 f (x) c2 c x (e) c1 x x c x y lim f (x) 5 lim f (x), so lim f (x) exists; f(c) is not defined y 5 f (x) c2 c x (c) c1 x x c x y lim f(x) 5 f(c) ? lim f(x), so lim f(x) does not exist; f(c) is defined y 5 f (x) c2 c x (f) c1 x x c Much earlier in this text, we stated that a function f is continuous if its graph could be drawn without lifting pencil from paper. Figure 10 reveals that the only graph that has this characteristic is the graph in Figure 10(a), for which the one-sided limits at c each exist and are equal to the value of f at c.This leads us to the following definition.
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