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Limits Question

  1. May 2, 2015 #1
    1. The problem statement, all variables and given/known data
    Calculate [tex]\lim_{(x,y)\to(0,0)}\frac{x^4-4y^2}{x^2+2y^2}[/tex] along the the line [tex]y=2x[/tex]

    2. Relevant equations
    N/A

    3. The attempt at a solution
    Not too sure what they mean by calculating the limit along the line [tex]y=2x[/tex]. The answer is [itex]\frac{-3}{5}[/itex].
    But I have gotten so far: [tex]\lim_{(0,y)\to(0,0)}\frac{-y^2}{y^2}=-1[/tex] and [tex]\lim_{(x,0)\to(0,0)}\frac{x^2}{x^2}=1[/tex], but the limit doesn't exist [tex]l_1\neq{l_2}[/tex]?
     
    Last edited by a moderator: May 2, 2015
  2. jcsd
  3. May 2, 2015 #2

    haruspex

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    Taking the limit along the line y=2x just means you can substitute y=2x then take the limit as x tends to 0.
    Your two attempts took the limits along the lines x=0 (first attempt), y=0 (2nd attempt).
     
  4. May 2, 2015 #3
    Sorry, it was supposed to be [tex]\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}[/tex], so you sub in [tex]y=2x[/tex] and compute [tex]\lim_{x\to0}\frac{x^2-(2x)^2}{x^2+(2x)^2}[/tex]?
     
  5. May 2, 2015 #4

    Mark44

    Staff: Mentor

    Yes.

    Side note: Don't use BBCodes inside of LaTeX code. Your BBCode italics tags broke ##l_1 \neq l_2## in your first post.
     
  6. May 2, 2015 #5

    SammyS

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    Were you able to evaluate this limit ?
     
  7. May 2, 2015 #6
    Yes.
     
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