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    Functions: Limits

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    Using the definition of a limit (rather than the limit theorems) prove that

    lim {x -> a+} f(x)

    exists and find the limit in each of the following cases

    a) f(x) = x/|x|, a = 0.

    b) f(x) = x + |x|, a = -1.

    c) f(x) = (x - 1)/(x^2 - 1), a = 1.

    In which cases do

    lim {x -> a-} f(x) and lim {x -> a} f(x)

    also exist?

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    https://brainmass.com/math/real-analysis/functions-limit-definitions-8447

    Solution Preview

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    First let me clearly state the definition of a limit in my words.

    Definition: The statement lim {x -> a} f(x) has the following precise definition. Given any real number >0, there exists another real number >0 so that if 0<|x-a|<, then |f(x)-L|<.

    Furthermore we can say the same for the two one sided limits.

    Definition: The statement lim {x -> a+} f(x) has the following precise definition. Given any real number >0, there exists another real number >0 so that if 0<|x-a+|<, then |f(x)-L|<.

    Definition: The statement lim {x -> a-} f(x) has the ...

    Solution Summary

    The existence of limits for functions is determined.

    $2.49

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