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Limit in multivariable calculus

  1. May 1, 2013 #1
    1. The problem statement, all variables and given/known data

    Examine lim (x,y) -> (0,0) of:

    [itex]\sqrt{x^{2}+1} - \sqrt{y^{2}-1}[/itex]

    [itex]\frac{\sqrt{x^{2}+1} - \sqrt{y^{2}-1}}{x^{2}+y^{2}}[/itex]

    3. The attempt at a solution

    Tried variable sub:

    [itex]\sqrt{x^{2}+1} = a, \sqrt{y^{2}-1} = b[/itex]

    [itex]\frac{a - b}{a^{2}-b^{2}}[/itex]

    (a -> 1, b -> 1 as x,y -> 0)

    Still nasty

    Tried polar coordinates:

    [itex]\sqrt{1 + r^{2}cosθ} - \sqrt{1 - r^{2}sinθ}/r^{2}[/itex]

    But I can't find a way for this limit to exist (which it is supposed to do according to the answers).
     
  2. jcsd
  3. May 1, 2013 #2
    I think I got it. Conjugate rule.....
     
  4. May 1, 2013 #3

    SammyS

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    Are you sure there's not some typo here?

    ##\displaystyle \sqrt{y^2-1\,}\ ## is only defined for |y| ≥ 1 .
     
  5. May 1, 2013 #4
    Yeah, meant to write 1-y^2. But I figured it out anyway :)
     
  6. May 1, 2013 #5
    Although I still have something I want to make clear: Say I wanted to show that a particular limit does NOT exist, for example:

    [itex]xy^{2}/(x^{2}+y^{4})[/itex]

    Is the following valid?

    Let y = x:

    [itex]x^{3}/(x^{2}+x^{4}) = x/1+x^{2}[/itex]

    Which is 0 as x,y -> 0

    Now let y = [itex]\sqrt{x}[/itex]

    [itex]x^{2}/(2x^{2})[/itex]

    Since I have found a path to the origin with an undefined limit, does that mean I have proven that the original limit does not exist?

    Sorry, I meant, is it sufficient to find two curves with DIFFERENT limits?

    I.e. let y = x and you have the limit = 0, and y = sqrt(x) and the limit = 1/2
     
    Last edited by a moderator: May 4, 2013
  7. May 1, 2013 #6

    SammyS

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    My first comment: Be careful with parentheses. You have unbalanced parentheses in one of your expressions. One of your other expression has a more serious problem. It needs an additional set of parentheses to make it true.
    What this says literally is that [itex]\ \displaystyle x^{3}/(x^{2}+x^{4}) = \frac{x}{1}+x^{2}\ .[/itex]

    What you meant to say (I hope) is [itex]\ \displaystyle x^{3}/(x^{2}+x^{4}) = x/\left(1+x^{2}\right)\,,\ [/itex] which is equivalent to [itex]\ \displaystyle x^{3}/(x^{2}+x^{4}) = \frac{x}{1+x^{2}}\ .[/itex]
    This should have been [itex]\ x^{2}/(2x^{2})\ .[/itex]

    If the limit along that second path was indeed undefined, then yes, that would show that the original limit does not exist.

    [itex]\displaystyle \lim_{x\to\,0} \left(\frac{x^{2}}{2x^{2}}\right)\ [/itex] does exist. What is it?

    (It's different than the limit along the path y = x. That does show that the original limit does not exist.)
     
  8. May 1, 2013 #7
    Yeah that was actually exactly what I meant but got confused in my latex-noobness :S
     
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