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Math Help - 2 calculus qestion

  1. #1
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    2 calculus qestion

    qestion 1
    If fsubx(xsub0, ysub0) and fy(xsub0, ysub0) both exist, then f is continuous at (xsub0, ysub0). Prove or disprove

    Hint consider the function defined by f(x,y) = (xy)/(x^2+y^2) , if (x,y) is not equal (0,0) or f(x,y) =0.

    question 2
    Let f(x,y) = (sin^2 (x-y))/ (abs(x) +abs(y) ).
    Prove that lim as (x,y) approach (0,0) of f(x,y)=0.

    Hint: for all real numbers m and n. abs(sin(m+n)) smaller than or equal to abo(m+n) smaller than or equal to abs(m) +abs(n).

    note: abs(m) means absolution value of m.

    Thank you very much
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  2. #2
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    Quote Originally Posted by littlemu View Post
    qestion 1
    If fsubx(xsub0, ysub0) and fy(xsub0, ysub0) both exist, then f is continuous at (xsub0, ysub0). Prove or disprove

    Hint consider the function defined by f(x,y) = (xy)/(x^2+y^2) , if (x,y) is not equal (0,0) or f(x,y) =0.
    False. Use the example the book gives.

    The function f(x,y) is defined on an open disk containing (0,0). Which means f is continous if and only if,
    lim [(x,y)-->(0,0)] f(x,y) = f(0,0)=0

    But if you choose the path x=y both approaching zero the limit of f(x,y) is then 1! It must always be zero if it continous.

    This is Mine 55th Post!!!
    Last edited by ThePerfectHacker; April 29th 2007 at 06:14 AM.
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  3. #3
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    I shall use my own inequality which is simple but useful.

    Let x,y>=0 then sqrt(x+y) <= sqrt(x)+sqrt(y)
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