UA796PRB - S olved Problems Introduction A simple...

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Solved Problems Introduction A simple modification of the standard limit operator, described in the next section is applied to a number of problems which would not be solvable with the standard definition of the limit. The solutions illustrate not only the viability of this approach to the calculus, but often reveal simpler, more direct, expressions for common structures in analysis. A brief introduction to the generalized limit We begin with Cauchy’s original definition, which is still used today: Definition (Cauchy, 1821) The number l is the limit of f(x) as x approaches a if for any positive number ε , however small, there is some positive number δ , which in general depends on ε and is such that | f ( x ) - l | < ε whenever 0 < | x - a | < δ . Suppose we modify this definition slightly by allowing the number l to vary with x , we would then have almost the same statement: Definition The function l ( x ) is the limit of f(x) as x approaches a if for any positive number ε , however small, there is some positive number δ , which in general depends on ε and is such that | f ( x ) - l ( x )| < ε whenever 0 < | x - a | < δ . This statement now tells us something about virtually any function, whether it converges or not: the functions f and l become more alike, the closer we get to the point a . We might call it a “functional limit” since the outcome of the limit operation is now a function, rather than a simple number. However, the usefulness of this definition is nullified by the large set Author: Chappell Brown First created: June , 1994 Last revision: June, 1996 Page: 1 © Copyright 1996 All rights reserved. Permission to copy and distribute for non-profit scholarly purposes granted. http://www.limit.com
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Solved Problems of possible outcomes. For example, every function could be its own limit since | f ( x ) - f ( x ) | = 0 < ε for all x . On the other hand, if we were to restrict the possible limit functions l ( x ) to only constant functions l ( x ) = c , we would essentially be back to the original definition of the limit. But by allowing special subsets of functions which exhibit typical divergent behavior to participate in the limit definition, it is possible to widen its domain of application while still preserving most of its familiar properties, including a single valued outcome. Also, in order to conform to the usual development of the calculus based on the limit operator, we need to somehow extract a numerical value from the function we obtain from this functional limit. These objectives can be realized by constructing a subset of functions from a limited set of basis functions. For example, suppose we define the set B as { l ( x ) | l ( x ) = b log (| x-a |) + c ; b , c real} Here B includes the constant functions, since we can choose b = 0, and it also includes a possible divergent term represented by log | x-a |, which becomes infinite as x tends to a . Now let us modify the definition of limit to read as follows Definition
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UA796PRB - S olved Problems Introduction A simple...

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