Chapter 2 | Limits
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Problem-Solving Strategy: Determining Continuity at a Point 1. Check to see if f ( a ) is defined. If f ( a ) is undefined, we need go no further. The function is not continuous at a . If f ( a ) is defined, continue to step 2. 2. Compute lim x → a f ( x ). In some cases, we may need to do this by first computing lim x → a − f ( x ) and lim x → a + f ( x ). If lim x → a f ( x ) does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If lim x → a f ( x ) exists, then continue to step 3. 3. Compare f ( a ) and lim x → a f ( x ). If lim x → a f ( x ) ≠ f ( a ), then the function is not continuous at a . If lim x → a f ( x ) = f ( a ), then the function is continuous at a . The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail. Example 2.26 Determining Continuity at a Point, Condition 1 Using the definition, determine whether the function f ( x ) = ( x 2 −4)/( x −2) is continuous at x =2. Justify the conclusion. Solution Let’s begin by trying to calculate f (2). We can see that f (2) =0/0, which is undefined. Therefore, f ( x ) = x 2 −4 x −2 is discontinuous at 2 because f (2) is undefined. The graph of f ( x ) is shown in Figure 2.35 .
Figure 2.35 The function f ( x ) is discontinuous at 2 because f (2) is undefined.
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