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Unformatted text preview: Example Before looking at a particular example, recall the formal definition of a limit : We say lim x → x f ( x ) = L if, for every number > 0, we can find a number δ > 0 so that < | x- x | < δ implies | f ( x )- L | < . The idea is that, given a number > 0, the value of δ > 0 is something over which you have control in order to constrain x via the inequality < | x- x | < δ, (1) and that we wish to make an appropriate selection of δ > 0 so that we can guarantee the desired or wanted inequality | f ( x )- L | < , (2) holds for this value of . The following steps can be used to verify a limit statement such as lim x → x f ( x ) = L. • Step #1: Rewrite the inequality (1) that we are able to control (i.e., the inequality involving δ ) as an interval around x- x , for x 6 = x . • Step #2: Rewrite the inequality (2) that we want to obtain (i.e., the inequality involving ) as an interval around x- x , for x 6 = x . This is often done as follows: (a) First, rewrite the inequality (2) that we want to obtain as an inequality centered...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
- Fall '10