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Find the largest δ that works .

  1. Jun 27, 2011 #1
    Find the largest δ that "works".

    1. The problem statement, all variables and given/known data
    Find the largest δ that "works" for the given ϵ:
    [itex]\displaystyle \lim_{x\to1}2x = 2; ϵ = 0.1[/itex]


    2. Relevant equations
    N/A


    3. The attempt at a solution
    Given ϵ > 0, then:
    if [itex]0 < |x - 1| < δ[/itex] then [itex]|2x - 2| < ϵ[/itex]
    ϵ = 0.1, so, [itex]|2x - 2| < 0.1[/itex]
    Now to establish the connection:
    [itex] |2x - 2| => |2 (x - 1)| => |2||x - 1| => 2|x - 1|[/itex]
    Therefore: [itex] 2|x - 1| < ϵ => |x - 1| < \frac{ϵ}{2} => |x - 1| <\frac{0.1}{2} =>|x - 1| < 0.05.[/itex] The largest value that "works" for δ is 0.05 since if:
    [itex] 0 < |x - 1| < 0.05[/itex] then [itex]|2x - 2| < 0.1[/itex]

    But, in my textbook, the answer is 2ϵ = 0.2 as the largest value that "works" for δ. So, I just wanted to know what I did wrong in my calculations as the book only shows the solution and not the work. Thanks in advance.
     
  2. jcsd
  3. Jun 27, 2011 #2

    Dick

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    Re: Find the largest δ that "works".

    I disagree with the textbook and I agree with you.
     
  4. Jun 27, 2011 #3
    Re: Find the largest δ that "works".

    I see. Thanks.

    I never would have thought university textbooks ever made such errors, even though high school ones were rife with them.
     
  5. Jun 27, 2011 #4

    Dick

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    Re: Find the largest δ that "works".

    The aren't that uncommon even at the university level. The same people are writing the textbooks.
     
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