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Determining Limits for Multivariable Functions

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data

    [tex]\lim_{\substack{x\rightarrow 2\\y\rightarrow 2}} f(x,y)=\frac{2x^2+2xy+2x-xy^2-y^3-y^2}{2x^3-2x^2y+2x-x^2y^2+xy^3-y^2}[/tex]

    2. Relevant equations


    3. The attempt at a solution

    Well, I tried using the line y = 2, and let x approach 2 (as well as making x=2 and let y approach 2), and nothing really seems to cancel when I do that.

    I also tried converting everything to polar form and let r approach 2[itex]\sqrt{2}[/itex], but again, everything just looks really messy.

    This is a previous year's test question, and it's not worth much marks, so I think I'm missing some trick/procedure to simplify this question.

    Also, I think this does not have a limit because the degree of the top (3) is less than the degree of the bottom (4); however, I'm not sure of how to prove this.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 16, 2011 #2


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    What do you mean by "nothing really seems to cancel". You should get the indeterminate form, 0/0 .

    Factoring the numerator gives [itex](x+y+1) (2 x-y^2)\,.[/itex]

    Factoring the denominator gives [itex](x^2-x y+1) (2 x-y^2)\,.[/itex]
  4. Oct 16, 2011 #3
    When I said "nothing seems to cancel", I just meant that the variables don't cancel out. For example, we can sometimes sub in y=mx^2, into a function to balance out the powers and allow the x to divide out, leaving a limit that changes depending on m.

    Anyway, thank you very much for helping me regarding this problem.
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