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Unformatted text preview: Jim Lambers Math 2A Winter Quarter 2003-04 Lecture 4 Notes These notes correspond to Section 2.3 in the text. Calculating Limits of Functions Now that we have precisely defined what the limit of a function is, we turn our attention to actually computing limits of certain functions. While we can always resort to the definition to compute the limit of a given function f ( x ) as x approaches a given value a , this is not the most practical course of action. Instead, it is best to use the definition to establish some laws that show how limits of more complicated functions can be obtained in terms of limits of simpler functions. Then, after learning how to compute limits for some very simple functions using the definition, we can use the laws to handle more complicated functions. We begin by learning how to compute limits of the simplest functions of all: the constant function f ( x ) = c , where c is any constant, and the identity function f ( x ) = x . For the constant function f ( x ) = c , the limit of f ( x ) as x approaches a must be equal to c . Since f ( x ) = c for all x , it is impossible for f to approach any other value. In summary, lim x → a c = c. Example Consider the constant function f ( x ) = 4, and suppose that we wish to compute its limit as x approaches 3. For any open interval I 1 containing 4, we can easily find an interval I 2 containing 3 such that f ( x ) is in I 1 for x in I 2 , because no matter what interval I 2 we choose, f ( x ) = 4 on I 2 , and therefore f ( x ) is in I 1 . It follows from the definition of a limit that f ( x ) must approach 4. a50 As for f ( x ) = x , as x approaches a , f ( x ) must also approach a , since f ( x ) = x . That is, lim x → a x = a. Proof Let I 1 be any interval containing a . According to the definition, lim x → a x = a if there is some interval I 2 containing a such that x is in I 1 whenever x is in I 2 . We can simply choose I 1 and I 2 to be the same interval, since they are both required to contain...
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