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Math Help - Norm Limits

  1. #1
    Mar 2012

    Norm Limits

    I hope this is in the right place, it feels like calculus, but it's the last part of my analysis problem.

    Construct an example where g: R2->R lim x->a g(x) exists but lim ||x||->||a|| g(x) does not exist

    I'm having a very hard time coming up with something to put this together. I think this is my theory behind it, does anyone have any ideas on something that would work?

    Specify a continuous functiong(x ⃗ )= g(x,y) on R^2, which is not constant, and which cannot be strictly written as a function of r = √(x^2+y^2 ). Then, the limit of g(x ⃗ ) as x ⃗ approaches a ⃗,a constant vector,will exist (and will equal g(a ⃗ ) ), but the limit of g(x ⃗ ) as |x ⃗ |approaches |a ⃗ | (a constant positive number) will not exist because x ⃗ can approach many different values in R2 (and still have|x ⃗ |approach |a ⃗ |), but the values that g(x ⃗ ) approach will be different.
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  2. #2
    MHF Contributor
    Sep 2012

    Re: Norm Limits

    Hey renelovexoxo.

    This kind of question reminds of results in complex analysis where only functions that have derivatives are analytic. You might want to get an example of a complex function in C that has no analytic derivative and use that to get one for your example.
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  3. #3
    Super Member
    Mar 2010

    Re: Norm Limits

    I think what you said is correct.

    In the first limit, x is approaching some point, but in the second, x is approaching the circle ||x||=||a||. So (assuming a is not the origin) it seems like any function that actually varies as you go around the circle would work.

    - Hollywood
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