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Implicit function theorem

  1. Nov 7, 2007 #1
    Hello all,

    I've been reading through my textbook about the Implicit function theorem and have come across something in one the examples that I just don't understand:

    Let F(x,y) = x - y^2 -1, then the partial der. F(a,b) WRT y = -2y, which is 0 only when b = 0. We know the possible solutions are y=+sqrt(x-1) or y=-sqrt(x-1).

    This is the part that I don't understand:

    "For x very close to a, only one of these solutions will be very close to b -- namely, sqrt(x-1) if b is greater than 0 and -sqrt(x-1) if b is less than 0 -- and this solution is the one that figues in the implicit function theorem."

    I'm not sure how they arrive at this conclusion for "x very close to a".

    Any help?

  2. jcsd
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