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Homework Help: Partial derivative and limits

  1. Dec 10, 2017 #1
    • Thread moved from the technical forums, so no Homework Template is shown
    Hello . I have problems with two exercises .
    1.[tex]\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} }[/tex]
    Here, I have to write when this limit will be exist.
    2.[tex]\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} }[/tex]
    Here, I have to write for which [tex] a \in \mathbb{R}_+[/tex] this limit will equal to zero.
    I don't have ideas how to do it.
  2. jcsd
  3. Dec 10, 2017 #2


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    Staff Emeritus
    Science Advisor

    Well, in a fraction, as the denominator approaches zero, then the fraction becomes undefined, unless the numerator also approaches zero. So under what circumstances does the numerator go to zero as [itex]t \rightarrow 0[/itex]?
  4. Dec 10, 2017 #3
    Yes. Now I know. When [tex] v_1=0[/tex] this limit will equal to zero.
  5. Dec 10, 2017 #4


    Staff: Mentor

    But the limit is as t approaches 0. As far as the limit process is concerned, ##v_1## is just some constant. You can't arbitrarily say it's zero.
  6. Dec 10, 2017 #5


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    Staff Emeritus
    Science Advisor

    The question was when (in what circumstances) the limit exists. When [itex]v_1 = 0[/itex] is a possible circumstance.
  7. Dec 10, 2017 #6

    Ray Vickson

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    Science Advisor
    Homework Helper

    For the second one, I would use polar coordinates ##h = r \cos \theta, k = r \sin \theta##, so that we are taking the limit as ##r \to 0##.
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