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Math Help - Show that f(x) = 1/x is continuous at x=5.

  1. #1
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    Show that f(x) = 1/x is continuous at x=5.

    Hi, I understand thus but I'm struggling with the proof.

    I have so far:

    f(5) = 1/5

    so we need to show that for all positive epsilon(E) such that that there exists positive delta(d) s.t |(1/x) - 1/5)| < E for all x satisfying |x-5|<d

    Then when I searched some threads on here I saw something that I think might help, but this is what it seemed to imply anyway :

    Now |(1/x) - (1/5)| = |x-5| / |x||5|
    However d>0 so when |x-5| = (1/2).|5| by the triangle law we would have
    |x|=> 5-|x-5| >(1/2).|5|

    therefore d <= (1/2) . |5|

    I don't really see where the bit in red came from, which is why I'm not sure what I have found is correct, and I am also unsure how to finish off the proof. Any help would be great thank you
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  2. #2
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    If |x-5|<1 then \left| {\frac{1}{{5x}}} \right| < 1.
    So let \delta  = \min \left\{ {1,\varepsilon } \right\}.
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