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Multivariable Calculus - a question of limits

  1. Mar 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove the following, using the meaning of a limit:

    2. Relevant equations
    epsilon > 0, delta > 0

    0 < sqrt(x^2 + y^2) < delta
    | f(x) - 0 | < epsilon (1)

    3. The attempt at a solution

    So, I know that I have to elaborate on the inequality in (1), further. However, I'm not sure of how to go about this. If I start transposing it seems to get much too messy and I end up with a proposed delta which seems far too complex. Is there some simplification that I'm overlooking here?

    Any help is really appreciated, ta.
  2. jcsd
  3. Mar 20, 2010 #2
    Could you not use polar coordinates instead?
    Then the equation would be much simpler and the limit is just as r goes to 0
  4. Mar 22, 2010 #3
    I did consider that but I'm pretty sure the limit definition must be used here (as a requirement).

    Does anyone have any ideas of how to simplify this beast?

    EDIT: Really need this one clarified guys. Does anyone have any hints at all? Even a vague idea.
  5. Mar 22, 2010 #4


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    Hint: use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).
  6. Mar 23, 2010 #5
    Bingo, great hint, thank you! Subtle yet it serves me well :D♦
    Last edited: Mar 23, 2010
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