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Applying the definition of a limit

  1. Aug 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Applying the definition of a limit to show that

    lim ((x^3 * y(y-1) ) / (x^2 + (y-1)^2) = 0 as (x,y) approaches (0,1)



    3. The attempt at a solution

    |x| = sqrt(x^2) <= sqrt((x^2 + (y-1)^2))
    |y-1|=sqrt((y-1)^2)<= sqrt((x^2 + (y-1)^2))
    |y|<= |y-1| + 1 via the triangle inequality

    Let e>0. We want to find d>0 such that

    0 < sqrt((x^2 + (y-1)^2)) < d then (|x|^3 |y| |y-1|)/ (x^2 + (y-1)^2) < e

    So

    (|x|^3 |y| |y-1|)/ (x^2 + (y-1)^2) <=
    [(sqrt((x^2 + (y-1)^2)))^3 * sqrt((x^2 + (y-1)^2)) * (sqrt((x^2 + (y-1)^2)) + 1) ] / (x^2 + (y-1)^2)

    Any ideas as to how to break down the right side of the inequality?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 18, 2013 #2

    vela

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    It's helpful to recognize that ##r = \sqrt{x^2 + (y-1)^2}## is the distance of the point (x,y) from the point (0,1). In terms of r, the righthand side is ##\frac{r^4(r+1)}{r^2} = r^2(r+1)##. Can you take it from there?
     
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