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Math Help - simple limits question

  1. #1
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    simple limits question

    Please have a look at this question:

    Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c

    The question is trivial to solve using the theorem that the limit of a polynomial in the set of complex numbers at a is P(a) or even using the theorems about the limits of f(x) + g(x) and limit of f(x)*g(x) where both f(x) and g(x) have limit at a.

    However I have to solve it using the epsilon delta definition of limit of complex functions i.e. given an epsilon, show that a delta can be constructed which would confine the mod of [f(x) - (a^2+c)] within the given epsilon. Despite several attempts, I have failed. Please can someone help?



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  2. #2
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    Quote Originally Posted by avicenna View Post
    Please have a look at this question:

    Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c

    The question is trivial to solve using the theorem that the limit of a polynomial in the set of complex numbers at a is P(a) or even using the theorems about the limits of f(x) + g(x) and limit of f(x)*g(x) where both f(x) and g(x) have limit at a.

    However I have to solve it using the epsilon delta definition of limit of complex functions i.e. given an epsilon, show that a delta can be constructed which would confine the mod of [f(x) - (a^2+c)] within the given epsilon. Despite several attempts, I have failed. Please can someone help?



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    If you had asked this only for real numbers then i could show the proof here. I have not read calculus in complex variable so i don't know.
    But in your text book there must be the proof of the fact that "limit of a polynomial in the set of complex numbers at a is P(a)". I guess the text books prove these theorems using epsilon-delta method for complex domain as well. I don't know if this helps but still....
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  3. #3
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    Thanks for your reply,

    Yes, Chrichill and Brown (the book I am using) does have proofs of the theorems you have mentioned in the set of complex numbers and indeed proving the given statement using those theorems is trivial. But I need to solve the question using the basic epsilon delta notation only, without the using the theorms on limits of polynomials or limits of functions multiplied/added together.

    I try to work backwards from what I am trying to show ( i.e. for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon), I use the triangle inequality a few times and arrive at a Delta. When I try to then work forward from that Delta to construct a step-by-step proof, that my Delta would indeed leed to the statement that I am trying to prove, it does not work. No matter how I construct the Delta (I have tried a few different ways), I cannot work forward to the statement:

    for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon

    Can anyone help, or otherwise convince me that I cannot do without employ the theorems of limits mentioned in the post.

    Thanks a lot
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  4. #4
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    Quote Originally Posted by avicenna View Post
    Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c
    Just take note that |(z^2+c)-(a^2+c)|=|z-a||z+a|.
    If we make sure that |z-a|<1 then we know |z+a|<1+2|a|.

    If \varepsilon  > 0 choose \delta  = \min \left\{ {1,\frac{\varepsilon }{{1 + 2\left| a \right|}}} \right\}.
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  5. #5
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by avicenna View Post
    Thanks for your reply,

    Yes, Chrichill and Brown (the book I am using) does have proofs of the theorems you have mentioned in the set of complex numbers and indeed proving the given statement using those theorems is trivial. But I need to solve the question using the basic epsilon delta notation only, without the using the theorms on limits of polynomials or limits of functions multiplied/added together.

    I try to work backwards from what I am trying to show ( i.e. for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon), I use the triangle inequality a few times and arrive at a Delta. When I try to then work forward from that Delta to construct a step-by-step proof, that my Delta would indeed leed to the statement that I am trying to prove, it does not work. No matter how I construct the Delta (I have tried a few different ways), I cannot work forward to the statement:

    for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon

    Can anyone help, or otherwise convince me that I cannot do without employ the theorems of limits mentioned in the post.

    Thanks a lot
    i checked out the book. in chapter 2 they have discussed the theorems on limits in section 12 of the book.
    The have proved it using epsilon-delta method. so you have to just read the proof and wherever they have written f(z) yo should read z^2+c and you will have proved the thing using epsilon delta. do you see what i am trying to say??
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  6. #6
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    I know what you mean and that would be easy to do. However, Plato has provided what I was looking for.
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  7. #7
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    Hi Plato,

    Thanks a lot for your help. I do not understand how you arrived at the third line?
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  8. #8
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    ok, got you. Thanks a million Plato.
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