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Math Help - Limits: Find Delta

  1. #1
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    Limits: Find Delta

    Hey, first post here and I just started this calculus I class.

    Here's the question:

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

    Find Delta such that if 0 < |x - 1| < Delta then |f(x) - 1| < 0.1

    Here's what I have so far:

    |f(x) - 1|
    = |2 - (1/x) - 1|
    = |1 - (1/x)| (my book shows |f(x) - 1| = |(1/x) - 1|)
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  2. #2
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    Re: Limits: Find Delta

    Quote Originally Posted by bearjew11 View Post
    Here's the question:
    f(x) = 2 - (1/x)
    Find Delta such that if 0 < |x - 1| < Delta then |f(x) - 1| < 0.1
    You must understand that |a-b|=|b-a|.
    Then \left| {\frac{1}{x} - 1} \right| = \frac{{\left| {x - 1} \right|}}{{\left| x \right|}}

    Now if |x-1|<0.5 then \frac{1}{2}<x<\frac{3}{2} or \frac{1}{{\left| x \right|}} < 2

    So pick 0 < \delta  < 0.05.
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  3. #3
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    Re: Limits: Find Delta

    Quote Originally Posted by bearjew11 View Post
    Hey, first post here and I just started this calculus I class.

    Here's the question:

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

    Find Delta such that if 0 < |x - 1| < Delta then |f(x) - 1| < 0.1

    Here's what I have so far:

    |f(x) - 1|
    = |2 - (1/x) - 1|
    = |1 - (1/x)| (my book shows |f(x) - 1| = |(1/x) - 1|)
    Yes that's the same thing: |a- b|= |b- a|
    Now you would want to write \frac{1}{x}- 1= \frac{1}{x}- \frac{x}{x}= \frac{1- x}{x} and \left|\frac{1- x}{x}\right|= \left|\frac{x- 1}{x}\right|.

    Now you can say "if |f(x)- 1|< \epsilon then \left|\frac{x- 1}{x}\right|< \epsilon and |x- 1|< |x|\epsilon so you only need an upper bound on |x|. If x is "close to 1", say between 0 and 2, what is the largest |x| can be? What about |x|\epsilon?
    Last edited by HallsofIvy; September 1st 2012 at 04:24 PM.
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