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Multivariable limits (NOT TO THE ORIGIN)

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi everyone! I'm pretty good with multivariable limits, but this one has me stumped:
    Find the limit or show that it does not exist:
    [tex]
    \underset{\left(x,y,z\right)\rightarrow\left(1,-1,1\right)}{\lim}\frac{yz+xz+xy}{1+xyz}
    [/tex]

    2. Relevant equations
    3. The attempt at a solution
    I could not work with polar coordinates here because there is no easy way to find a value that ρ approaches (the point is (1, -1, 1).
    I found it difficult to prove that the limit didn't exist, as in this case the point is in 4-d space and one can only approach it from various 3-D spaces, which I simply could not visualize how to do.
    Help please!
     
    Last edited: Mar 19, 2012
  2. jcsd
  3. Mar 19, 2012 #2
    simply take the limit along one axis, say x axis, the limit already blows up
     
  4. Mar 19, 2012 #3
    Read the problem buddy. The limit is to be taken approaching the point (1, -1, 1), which is a point that doesn't lie on any of the axes.
     
  5. Mar 19, 2012 #4

    Dick

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    The numerator approaches 0 and the denominator doesn't approach 0. Isn't that enough to tell you about the behavior of the limit?
     
  6. Mar 19, 2012 #5
    Doesn't the denominator approach: [itex] \left(1+\left(1\times-1\times1\right)\right)=\left(1+\left(-1\right)\right)=\left(1-1\right)=0 [/itex]?

    sunjin09, I realize I was a little rude with my response and I'm sorry, it's been a tough week. Thanks for the responses guys! I appreciate it.

    Quick note: plugging into Wolfram Alpha yields nothing, so I don't have an answer we can cross-reference against. Sorry!
     
  7. Mar 19, 2012 #6

    HallsofIvy

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    Yes, the denominator approaches 0. And the numerator doesn't. That was Dick's point.
     
  8. Mar 19, 2012 #7

    Dick

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    Yes, I had them backwards, sorry.
     
  9. Mar 19, 2012 #8
    Oh my god, the limit is -∞? I'm so sorry guys, I feel like a real idiot. Thanks for the help!
     
  10. Mar 19, 2012 #9

    Dick

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    You are welcome! But I wouldn't describe it that way. The denominator doesn't have a definite sign. It could be either +∞ or -∞ depending on how you approach it.
     
  11. Mar 20, 2012 #10
    No worries. I should've been more accurate in my wording.
     
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