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Math Help - Delta epsilon proof of a continuous limit

  1. #1
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    Delta epsilon proof of a continuous limit

    Hi I was just wondering for some help on proofing that this limit exists with the δ-ε proof


    Lim (x,y) -> (-1,0). X(x-1)y^2 / (x-1)^2 + y^2

    thankyou, I have tried it, but not sure if doing it right
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  2. #2
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    Re: Delta epsilon proof of a continuous limit

    To prove \displaystyle \lim_{(x,y) \to (-1, 0)} \frac{x(x-1)y^2}{(x-1)^2 + y^2} = 0 you need to show \displaystyle \sqrt{ (x + 1)^2 + y^2 } < \delta \implies \left| \frac{x(x-1)y^2}{(x-1)^2 + y^2} - 0 \right| < \epsilon

    Can you show us what you've tried please?
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  3. #3
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    Re: Delta epsilon proof of a continuous limit

    what was done above and ->

    y^2 <(or equal to) (x+1)^2 + y^2 since x > -1

    so, lx(x-1)l y^2 / (x-1)^2 + y^2 <(or equal too) lx(x+1)l = sqrt((x(x+1))^2) <(or equal too) sqrt((x+1)^2+ y^2)

    therefore, δ=ε

    Hence, Lim (x,y) -> (-1,0). X(x-1)y^2 / (x-1)^2 + y^2 = 0

    is that right?

    thankyou
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