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Proving limits with epsilon and delta

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that as x approaches 0, sin(1/x) has no limit.

    2. Relevant equations

    |x-a|<d and f(x)-L<e

    3. The attempt at a solution

    my teacher explained it, but i didnt quite get where the contradiction is at the end. We chose epsilon to be 1/2
     
  2. jcsd
  3. Feb 23, 2010 #2
    Following your teacher's hint, assume that
    [tex]\lim_{x\to 0} \sin(1/x) = L.[/tex]
    Then for every ε>0, there exists a η>0 such that 0<|x-0|<η implies |sin(1/x)-L|<ε. Now, let ε=1/2. Then there exists a δ>0 such that 0<|x|<δ implies |sin(1/x)-L|<1/2.

    Now that I've started it, can you keep it going until you get a contradiction?
     
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