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Math Help - Analysis question

  1. #1
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    Analysis question

    suppose that f : (a,b)\{c} ----> real numbers is a function such that

    lim (x--->c+) {f(x)} and lim (x--->c-) {f(x)} both exist and are equal to a common value l.

    how can we prove that lim (x--->c) {f(x)} exists and that it equals l?
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  2. #2
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    If \varepsilon  > 0 and 0<|x-c|<\varepsilon then one of these is true, x\in (c-\varepsilon ,c)\text{ or }x\in (c,c+\varepsilon).
    The first is involved with the limit from the left the other the limit from the right.
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  3. #3
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    Let  \epsilon > 0 be given. By assumption, we can find a  \delta_1, \delta_2 such that for  x \in (c - \delta_1, c) or  x \in (c, c + \delta_2, we have that  |f(x) - 1| < \epsilon .

    Set  \delta = min(\delta_1, \delta_2) . Then for  x \in (c - \delta, c+\delta), x \ne c , it follows that  |f(x) - 1| < \epsilon .
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