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Math Help - prove or disprove... involving sided limits

  1. #1
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    prove or disprove... involving sided limits

    How can I prove the following:
    \lim_{x\to 0}f(x)=L implies \lim_{x\to \infty}f(\frac{1}{x})=L
    For that matter, how would I disprove this:
    \lim_{x\to \infty}f(\frac{1}{x})=L implies \lim_{x\to 0}f(x)=L
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    How can I prove the following:
    \lim_{x\to 0}f(x)=L implies \lim_{x\to \infty}f(\frac{1}{x})=L
    For that matter, how would I disprove this:
    \lim_{x\to \infty}f(\frac{1}{x})=L implies \lim_{x\to 0}f(x)=L
    \forall~\epsilon>0, \exists~\delta>0 such that |y-0|=|y|<\delta \implies |f(y)-L|<\epsilon

    Replace y with \frac{1}{x} and we have |1/x|<\delta \implies |f(1/x)-L|<\epsilon.

    So \forall~\epsilon>0, \exists~\delta>0 such that x>1/\delta \implies |f(1/x)-L|.

    This means \lim_{x\to\infty}f(1/x)=L.

    -------------

    I cannot think of a counterexample for the second one. I'm sure someone else will, though.
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