956 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function In Words A rational function is continuous at every number in its domain. DEFINITION A function f is continuous at c if and only if • f is defined at c ; that is, c is in the domain of f so that f c( ) equals a number. • f x f c lim x c ( ) ( ) = → − • f x f c lim x c ( ) ( ) = → + In other words, a function f is continuous at c if and only if f x f c lim x c ( ) ( ) = → If f is not continuous at c , we say that f is discontinuous at c . Each function whose graph appears in Figures 10(b) to 10(f) is discontinuous at c . Now Work PROBLEM 27 Look again at the Limit of a Polynomial theorem on page 949. Based on the theorem, we conclude that a polynomial function is continuous at every number. Look at the Limit of a Quotient theorem on page 950 and suppose f and g are polynomial functions.We conclude that a rational function is continuous at every number, except any numbers at which it is not defined. At numbers where a rational function is not defined, either a hole appears in the graph or else a vertical asymptote appears. Determining the Numbers at Which a Rational Function Is Continuous (a) Determine the numbers at which the rational function R x x x x 2 6 8 2 ( ) = − − + is continuous. (b) Use limits to analyze the graph of R near 2 and near 4. (c) Graph R . Solution EXAMPLE 2 (a) Since R x x x x 2 2 4 , ( ) ( )( ) = − − − the domain of R is x x x 2, 4 . { } ≠ ≠ R is a rational function and it is defined at every number except 2 and 4. We conclude that R is continuous at every number except 2 and 4, since 2 and 4 are not in the domain of R . (b) To analyze the behavior of the graph of R near 2 and near 4, look at R x lim x 2 ( ) → and R x lim . x 4 ( ) → • For R x lim , x 2 ( ) → we have R x x x x x lim lim 2 2 4 lim 1 4 1 2 x x x 2 2 2 ( ) ( )( ) = − − − = − = − → → → As x gets closer to 2, the graph of R gets closer to 1 2 . − Since R is not defined at 2, the graph will have a hole at 2, 1 2 . ( ) − • For R x lim , x 4 ( ) → we have R x x x x x lim lim 2 2 4 lim 1 4 x x x 4 4 4 ( ) ( )( ) = − − − = − → → → If x 4 < and x gets closer to 4, the value of x 1 4 − is negative and becomes unbounded; that is, R x lim . x 4 ( ) = −∞ → −
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