Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is my question. Let $p$ be a limit point of a set $ A \subseteq \mathbb{R}^k$. Let $f: (-\delta, \delta) \rightarrow \mathbb{R}^k $ be a function which is differentiable in $0$, $\delta > 0$, for $t>0$, $f(t) \in A$ and $f(0)=p$. Thesis: $f^{'}(0) $ is tangent to $A$ in point $p$.

Well, the definition of a tangent vector $v$ is: $v = t\lim_{n \to \infty} \frac{p_n-p}{||p_n-p||} $ (where $p_n \to p$) and I think this could be done by definition, but how? Please, help me.

share|improve this question

1 Answer 1

If you call $v = f'(0)$ then you have $$ v = \lim_{t \to 0^+} \frac{f(t)-p}{t} = \lim_{t \to 0^+} \frac{f(t)-p}{\|f(t)-p\|}\Bigl\|\frac{f(t)-p}{t}\Bigr\| = |v|\lim_{t \to 0} \frac{f(t)-p}{\|f(t)-p\|} $$ You can choose any sequence $t_n$ that tends to $0$ and then $p_n = f(t_n)$.

share|improve this answer
1  
+1 No ill will...Besides, our posts were/are good! ;) –  amWhy Jan 26 '13 at 0:47
    
@Guillermo: I have been on the other end of a similar situation, where my answer, arrived at independently, looked very similar to another answer that was posted just before mine. When answers are very close, voters will often go with the earlier answer. If a pattern of abuse forms (which I seriously doubt), flag us again. –  robjohn Jan 26 '13 at 0:49
    
Why do we write $t$ approaching to $0$ from the right side? –  Anne Feb 9 '13 at 19:40
    
so that when I write $\|(f(t)-p)/t\|$ I don't have to deal with the sign of $t$. –  Guillermo Feb 11 '13 at 3:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.