How to go from limit of vector norm to 'normal' limit

1. Dec 11, 2012

Berrius

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 $$f:U \subseteq R^n -> R^m$$ be a function which is differentiable at $$a \in U$$, and $$u \in R^n$$
It is then stated that it is clear that:
$$lim_{t \to 0} \frac{||f(a+t*u)-f(a)-D_f(t*u)||}{||t*u||} = 0 => lim_{t \to 0} \frac{f(a+t*u)-f(a)-D_f(t*u)}{|t|} = 0$$

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. Dec 11, 2012

Staff: Mentor

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.

3. Dec 11, 2012

Berrius

I know ofcourse ||v||=0 iff v=0, but why is this still true when i put a limit in front of it?

4. Dec 11, 2012

Staff: Mentor

You can show it for each component in some basis, for example. They all go to 0.

5. Dec 11, 2012

Vargo

It is the definition that vector v converges to vector w if the norm of their difference converges to 0.

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