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2-variable function limit

  1. Nov 24, 2011 #1
    [itex]lim_{(x,y)\rightarrow(0,0)}\frac{y^{3}}{x^{4}+sin^{2}y}[/itex]

    I need help with this above

    any suggestions?

    i dont know how to do this and have exam tomorrow :(
     
  2. jcsd
  3. Nov 24, 2011 #2
    please im short of time......

    i would be very grateful
     
  4. Nov 24, 2011 #3
    I'll give you my take on this

    Note that

    [tex]x^4 \geq 0 [/tex]

    And that

    [tex]0\leq sin^2 y \leq 1[/tex]

    Then we know that

    [tex]x^4 \leq sin^2 y + x^4 \leq 1 + x^4[/tex]

    Also,all of this is just to tell you that you may have to use the squeeze theorem to solve this.
     
  5. Nov 24, 2011 #4

    LCKurtz

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    Try a couple of different paths, like letting y → 0 first or x → 0 first to see if perchance they are different.
     
  6. Nov 24, 2011 #5
    thanks for a tip, but I've already figured this and it probably doesnt take me any step further :(

    Do you mean calculating [itex]lim_{x\rightarrow0}(lim_{y\rightarrow0}A)[/itex] , where A =

    [itex]\frac{y^{3}}{x^{4}+sin^{2}y}[/itex]
     
  7. Nov 24, 2011 #6

    LCKurtz

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    Yes. And the reverse order too. What can you conclude if they come out not equal to each other?
     
  8. Nov 24, 2011 #7
    That the limit as (x,y)->(0,0) does not exist?

    but i dont know how to calculate these limit (d'hospital) ??
     
  9. Nov 24, 2011 #8

    LCKurtz

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    That's correct.
    Have you tried anything? What happens if y → 0 first? What happens if x → 0 first?
     
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