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Math Help - Delta-Epsilon proof

  1. #1
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    Delta-Epsilon proof

    I am having a real tough time with these proofs.

    f(x)=2-(1/x)

    find delta such that 0<|x-1|<d then |f(x)-1|<0.1

    I understand what it means, but I'm having a hard time proving it.

    I start out by working the epsilon inequality...

    I end up with something like this...
    |1/x||x-1|<0.1

    Now I'm stuck. But from what I've been reading you would set
    |x-1|<1

    This is obviously not a good option given 1/x. So I chose 1/2.
    I eventually get...
    1/2<x<3/2

    Now I don't understand the next step. This should be so easy.
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  2. #2
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    Let \delta = .05 and \left| {x - 1} \right| < \delta < \frac{1}{2}.
    From what you have already done this means that \frac{1}{x} < 2.
    Thus \left| {1 - \frac{1}<br />
{x}} \right| = \left| {\frac{{x - 1}}<br />
{x}} \right| < 2\left| {x - 1} \right| < 2\left( \delta \right) = 0.1
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