Chap2_Sec2

# Chap2_Sec2 - LIMITS LIMITS 2 2.2 The Limit of a Function...

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Unformatted text preview: LIMITS LIMITS 2 2.2 The Limit of a Function LIMITS In this section, we will learn: About limits in general and about numerical and graphical methods for computing them. Let’s investigate the behavior of the function f defined by f ( x ) = x 2 – x + 2 for values of x near 2. The following table gives values of f ( x ) for values of x close to 2, but not equal to 2. THE LIMIT OF A FUNCTION From the table and the graph of f (a parabola) shown in the figure, we see that, when x is close to 2 (on either side of 2), f ( x ) is close to 4. THE LIMIT OF A FUNCTION In fact, it appears that we can make the values of f ( x ) as close as we like to 4 by taking x sufficiently close to 2. THE LIMIT OF A FUNCTION We express this by saying “the limit of the function f ( x ) = x 2 – x + 2 as x approaches 2 is equal to 4.” The notation for this is: ( 29 2 2 lim 2 4 x x x →- + = THE LIMIT OF A FUNCTION In general, we use the following notation. We write and say “the limit of f ( x ), as x approaches a, equals L ” if we can make the values of f ( x ) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a ) but not equal to a . ( 29 lim x a f x L → = THE LIMIT OF A FUNCTION Definition 1 Roughly speaking, this says that the values of f ( x ) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a ) but x a . A more precise definition will be given in Section 2.4. THE LIMIT OF A FUNCTION ≠ An alternative notation for is as which is usually read “ f ( x ) approaches L as x approaches a .” ( 29 lim x a f x L → = THE LIMIT OF A FUNCTION ( ) f x L → x a → Notice the phrase “but x a ” in the definition of limit. This means that, in finding the limit of f ( x ) as x approaches a , we never consider x = a . In fact, f ( x ) need not even be defined when x = a . The only thing that matters is how f is defined near a . THE LIMIT OF A FUNCTION ≠ The figure shows the graphs of three functions. Note that, in the third graph, f ( a ) is not defined and, in the second graph, . However, in each case, regardless of what happens at a , it is true that . THE LIMIT OF A FUNCTION ( ) f x L ≠ lim ( ) x a f x L → = 2 1 1 lim 1 x x x →-- THE LIMIT OF A FUNCTION Example 1 lim ( ) x a f x → Guess the value of . The tables give values of f ( x ) (correct to six decimal places) for values of x that approach 1 (but are not equal to 1). On the basis of the values, we make the guess that THE LIMIT OF A FUNCTION Example 1 2 1 1 lim 0.5 1 x x x →- =- Example 1 is illustrated by the graph of f in the figure....
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Chap2_Sec2 - LIMITS LIMITS 2 2.2 The Limit of a Function...

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