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Limit of Functions

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Negate the definition of the limit of a function, and use it to prove that for the function
    f : (0; 1) --> R where f(x) 1/x, lim x-->0 f(x) does not exist.

    2. Relevant equations

    The limit of f at a exists if there exists a real number L in R such that for every e>0 there exists d>0 such that for every x in the interval with 0<|x-a|<d then |f(x)-L|<e.

    3. The attempt at a solution

    The limit of f at a does not exist if for all real numbers L in R such that for every e>0 there exists d>0 such that there exists xin the interval with 00<|x-a|<d then |f(x)-L|>e
     
  2. jcsd
  3. Apr 22, 2012 #2

    SammyS

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    The following is true even if L is the limit at a .


    For every δ > 0 there needs to be an ε > 0 (this ε usually depends upon the δ) such that there is some x0 for which 0 < |x0-a| < δ and |f(x0)-L|> ε .
     
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