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Epsilon and Delta Proof

  1. Sep 20, 2010 #1
    1. The problem statement, all variables and given/known data
    I am currently having problems with a similar question, and used that post, but I'm finding it hard to solve for x.

    the question states. if f(x) = 1/x for every x > 0, there is a positive quantity e (epsilon), find the d(delta) quatity such that

    if 0 < l x - 3 l < d, then l 1/x - 1/3 l < e

    i simplify l 1/x - 1/3 l < e to l (x - 3)/ 3x l < e, i put the x - 3 in brackets to show that is all over the 3x

    then show the relation between d = e, which is if l x - 3 l < d, and l (x - 3)/3x l < e
    then l d/3x l = e.

    I think I am almost there just need a couple pointers in the right direction i believe.
     
  2. jcsd
  3. Sep 20, 2010 #2
    Use the fact that x is positive to show that l (x - 3)/3x l = |x - 3|/3x < d/3x. Note that d should be defined by an inequality (a continuous range of values for each e), not an equality.
    These types of proofs are usually done backwards.
    Use a scratch-pad to note that if we want the expression to be bounded by e and the expression is necessarily bounded by d/3x, then we want d/3x < e, which implies d < (some expression involving e).
    On the real proof, you then magically say "Let d < (that expression involving e)." and then proceed to show that the expression satisfies the inequality.
     
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