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Math Help - Epsilon Delta Proof of a Limit

  1. #1
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    Epsilon Delta Proof of a Limit

    The problem is give a rigerous epsilon delta proof that lim (x,y) (0,0) (x^3 + y^3)/(x^2 + y^2)

    Heres what I have. Given any ε > 0, we can find a corresponding δ > 0 such that if 0 < || (x,y) - (0,0)|| < δ , then (x^3 + y^3)/(x^2 + y^2) < ε

    0 < √(x^2 +y^2) < δ

    √(x^2) =|x|< δ
    √(y^2) = |y| < δ

    I'm stuck after this part. Is the above part even correct? If so, how do I finish i then?

    Thanks
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  2. #2
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    Quote Originally Posted by meks08999 View Post
    The problem is give a rigerous epsilon delta proof that lim (x,y) (0,0) (x^3 + y^3)/(x^2 + y^2)

    Heres what I have. Given any ε > 0, we can find a corresponding δ > 0 such that if 0 < || (x,y) - (0,0)|| < δ , then (x^3 + y^3)/(x^2 + y^2) < ε

    0 < √(x^2 +y^2) < δ

    √(x^2) =|x|< δ
    √(y^2) = |y| < δ

    I'm stuck after this part. Is the above part even correct? If so, how do I finish i then?

    Thanks
    \left|\frac{x^3+y^3}{x^2+y^2} \right| =\frac{|x^3+y^3|}{x^2+y^2} \leq \frac{x^2|x| + y^2|y|}{x^2+y^2} \leq |x| + |y| < 2 \delta. so we need to choose \delta = \frac{\epsilon}{2}.
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  3. #3
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    Thanks for the help.
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