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Math Help - limits and continuity

  1. #1
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    limits and continuity

    1a)Do these functions have limits.If the limit exists, find it with justification, if not explain why not


    i) f(x,y) =x-y/x+y
    ii) f(x,y) =x-y/x+y
    iii) f(x,y) =xy/|x|+|y|
    iv) f(x,y) =1-√(1-x)/x+ xy+y
    v) f(x,y) =yx/y^6+x

    im having problems finding the limits especially iii) and iv). i know that you can prove a limit doesn't exist by demonstration but how do you justify it if it does exist

    b) Let f be a real valued function of two real variables defined by:

    f(x,y) ={2x sin y/ x+y if y >x
    sin(x+y) if y≤x
    Determine the points at which f is continuous.
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  2. #2
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    Let's look at (iii). I claim the limit is zero: that is, for all epsilon > 0 there exists delta > 0 such that if 0 < sqrt(x^2+y^2) < epsilon then | xy(|x|+|y|) | < delta.
    I claim we can take delta = epsilon. If 0 < sqrt(x^2+y^2)<epsilon, then |x| and |y| are less than epsilon and one of them, say x, is non-zero. Then |x|.|y|/(|x|+|y| < epsilon. |x|/(|x| + epsilon) < epsilon as claimed.
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