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Homework Help: How to go from limit of vector norm to 'normal' limit

  1. Dec 11, 2012 #1
    This is not really a homework question, but i've come across this while preparing for a test

    1. The problem statement, all variables and given/known data
    Let [tex]f:U \subseteq R^n -> R^m[/tex] be a function which is differentiable at [tex]a \in U[/tex], and [tex]u \in R^n[/tex]
    It is then stated that it is clear that:
    [tex] lim_{t \to 0} \frac{||f(a+t*u)-f(a)-D_f(t*u)||}{||t*u||} = 0 => lim_{t \to 0} \frac{f(a+t*u)-f(a)-D_f(t*u)}{|t|} = 0[/tex]

    How do they get this result?

    3. The attempt at a solution
    I've tried using the epsilon/delta-def of limits but where the first limit is about real numbers, the second is about vectors in the R^m, so im totally confused.
     
  2. jcsd
  3. Dec 11, 2012 #2

    mfb

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    Denominator: ##||t \cdot u|| = |t| \cdot ||u||## and the limit is linear in ##||u||##.
    Numerator: The norm of a vector can go to 0 if and only if the vector itself goes to 0.
     
  4. Dec 11, 2012 #3
    I know ofcourse ||v||=0 iff v=0, but why is this still true when i put a limit in front of it?
     
  5. Dec 11, 2012 #4

    mfb

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    You can show it for each component in some basis, for example. They all go to 0.
     
  6. Dec 11, 2012 #5
    It is the definition that vector v converges to vector w if the norm of their difference converges to 0.
     
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