954 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function is called the left-hand limit . It is read “the limit of f x( ) as x approaches c from the left equals L ” and may be described by the following statement: As x gets closer to c , but remains less than c , the corresponding value of f gets closer to L . As x gets closer to c, but remains greater than c, the corresponding value of f gets closer to R . The notation → − x c is used to remind us that x is less than c . The notation f x R lim x c ( ) = → + is called the right-hand limit . It is read “the limit of f x( ) as x approaches c from the right equals R ” and may be described by the following statement: The notation → + x c is used to remind us that x is greater than c . Figure 7 illustrates left-hand and right-hand limits. Figure 7 Left and right limits x x y lim R c x (b) x x y lim L c x (a) Figure 8 x y lim f (x) 5 lim f (x) L 5 R c2 c x (a) c1 x x y lim f (x) ? lim f (x) R c2 c x (b) c1 x L THEOREM The limit L of a function y f x( ) = as x approaches a number c exists if and only if both one-sided limits exist at c and are equal. That is, f x L f x f x L lim ifandonlyif lim lim x c x c x c ( ) ( ) ( ) = = = → → → − + The left-hand and right-hand limits can be used to determine whether f x lim x c ( ) → exists. See Figure 8. As Figure 8(a) illustrates, f x lim x c ( ) → exists and equals the common value of the left-hand limit and the right-hand limit L R . ( ) = In Figure 8(b), we see that f x lim x c ( ) → does not exist because L R. ≠ This leads to the following result. Collectively, the left-hand and right-hand limits of a function are called one-sided limits of the function. In Words x c → − means x is approaching c from the left, so x c. < In Words x c → + means x is approaching c from the right, so x c. >
RkJQdWJsaXNoZXIy NjM5ODQ=