lecture3_1 - 2.3 The Precise Definition of a Limit Example...

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Unformatted text preview: 2.3 The Precise Definition of a Limit Example 1 Consider the function near . Intuitively it is clear that y is close to when x is close to 4, so However, how close to x = 4 does x have to be so that y = 2 x- 1 differs from 7 by, say, less than 2 units? 1 Using the Absolute Value | x | < a ⇔ | x | > a ⇔ a < 1 x < b ⇔ x 2 < a ⇔ a < x 2 ⇔ Odd and Even Functions A function f ( x ) is even , if f ( x ) = f (- x ). The graph of an even function is symmetric across the y-axis. Example: A function f ( x ) is odd , if f ( x ) =- f (- x ). Example: 2 Definition of Limit Let f ( x ) be defined on an open interval about x , except possibly at x itself. We say that the limit of f ( x ) as x approaches x is the number L , and write if, for every number > 0, there exists a corresponding number δ > such that for all x , 3 Example 2 Show that lim x → 1 (5 x- 3) = 2. Solution Set x = , f ( x ) = , and L = in the definition of limit....
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lecture3_1 - 2.3 The Precise Definition of a Limit Example...

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