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Norm Limits

  1. Apr 23, 2013 #1
    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.
  2. jcsd
  3. Apr 24, 2013 #2


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    2017 Award

    Staff: Mentor

    Apart from constant functions, basically everything not too complicated works.
    Write down the easiest non-constant function you can imagine, chances are good that it is an example you can use.
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