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Proving that a limit is non-existent

  1. May 8, 2012 #1
    1. The problem statement, all variables and given/known data
    The function is:
    f: D={(x,y)[itex]\in[/itex]ℝ2:x+y≠0}→ℝ
    (x,y)→[itex]\frac{x-y}{x+y}[/itex]

    They ask you to prove that the limit as (x,y)→(0,0) is non-existent.

    2. Relevant equations



    3. The attempt at a solution

    My attempt at a solution was using the definition of limit: If there was a limit (L[itex]\in[/itex]ℝ) when (x,y)→(0,0), then [itex]\forall[/itex]ε>0 there would be a δ>0 for which:

    ||(x,y)||<δ →(implied) |f(x,y)-L|<ε

    I tried guessing an ε for which there was no δ, hence proving the non-existence of the limit, but I can't seem to find it.

    Another attempt at a solution was changing the expression to polar coordinates which gave me the following:

    f(r,θ)=[itex]\frac{cosθ-senθ}{cosθ+senθ}[/itex] [itex]\forall[/itex]r>0

    The limit when r→0 is always the same: f(r,θ). But since the function oscillates indeterminately, there's no limit.

    Am I doing this right? I really need some good feedback on this ;)
     
  2. jcsd
  3. May 8, 2012 #2
    You have some interesting ideas. My simple version would be to note that if you approach (0,0) along the two axes, you get different limits.
     
  4. May 8, 2012 #3

    HallsofIvy

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    It should be sufficient to point out that there exist points (x, y), on the line y= x, arbitrarily close to (0, 0) such that f(x, y)= 0 and, on the line y= -x, arbtririly close to (0, 0), for which f(x, y) is not defined.
     
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