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2.2 Limit1 - 88 | C H A P T E R 2 L I M I T S A N D D E R I...

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88 |||| CHAPTER 2 LIMITS AND DERIVATIVES THE LIMIT OF A FUNCTION Having seen in the preceding section how limits arise when we want to find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them. Let’s investigate the behavior of the function defined by for val- ues of near 2. The following table gives values of for values of close to 2, but not equal to 2. From the table and the graph of (a parabola) shown in Figure 1 we see that when is close to 2 (on either side of 2), is close to 4. In fact, it appears that we can make the values of as close as we like to 4 by taking sufficiently close to 2. We express this by saying “the limit of the function as approaches 2 is equal to 4.” The notation for this is In general, we use the following notation. DEFINITION We write and say “the limit of , as approaches , equals ” if we can make the values of arbitrarily close to (as close to L as we like) by taking x to be sufficiently close to (on either side of ) but not equal to . Roughly speaking, this says that the values of tend to get closer and closer to the number as gets closer and closer to the number (from either side of ) but . (A more precise definition will be given in Section 2.4.) An alternative notation for is as which is usually read “ approaches as approaches .” a x L f x x l a f x l L lim x l a f x L x a a a x L f x a a a L f x L a x f x lim x l a f x L 1 lim x l 2 x 2 x 2 4 x f x x 2 x 2 x f x f x x f x f x x f x x 2 x 2 f 2.2 4 ƒ approaches 4. x y 2 As x approaches 2, y= -x+2 0 FIGURE 1 x 3.0 8.000000 2.5 5.750000 2.2 4.640000 2.1 4.310000 2.05 4.152500 2.01 4.030100 2.005 4.015025 2.001 4.003001 f x x 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 f x
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Notice the phrase “but ” in the definition of limit. This means that in finding the limit of as approaches , we never consider . In fact, need not even be defined when . The only thing that matters is how is defined near . Figure 2 shows the graphs of three functions. Note that in part (c), is not defined and in part (b), . But in each case, regardless of what happens at , it is true that . EXAMPLE 1 Guess the value of . SOLUTION Notice that the function is not defined when , but that doesn’t matter because the definition of says that we consider values of that are close to but not equal to . The tables at the left give values of (correct to six decimal places) for values of that approach 1 (but are not equal to 1). On the basis of the values in the tables, we make the guess that M Example 1 is illustrated by the graph of in Figure 3. Now let’s change slightly by giving it the value 2 when and calling the resulting function : This new function still has the same limit as approaches 1 (see Figure 4). 0 1 0.5 x-1 -1 y= FIGURE 3 FIGURE 4 0 1 0.5 y=© 2 y x y x x t t ( x ) x 1 x 2 1 if x 1 2 if x 1 t x 1 f f lim x l 1 x 1 x 2 1 0.5 x f x a a x lim x l a f x x 1 f x x 1 x 2 1 lim x l 1 x 1 x 2 1 (c) x y 0 L a (b) x y 0 L a (a) x y 0 L a FIGURE 2 lim ƒ=L in all three cases x a lim x l a f x L a f a L f a a f x a f x x a a x f x x a SECTION 2.2 THE LIMIT OF A FUNCTION |||| 89 0.5 0.666667 0.9 0.526316 0.99 0.502513 0.999
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