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We have a function: $\phi : U \to \mathbb{R}^l$, where $U \subseteq \mathbb{R}^k$ is an open subset. $\phi$ is a function of class $C^1$, differentiable in $q$, $q \in U$. Let $\displaystyle q_n \to q$ and $\displaystyle \phi(q_n) \to \phi(q)$ while $\displaystyle n \to \infty$. How to prove that: $$ \lim_{n \to \infty} \frac{D\phi (q) (q_n - q)}{||q_n -q||} = \lim_{n \to \infty} \frac{\phi (q_n) - \phi(q)}{||q_n -q||} \text{?} $$ There is a hint to use the definition of $D\phi(q)$ and a fact that it is a linear map. I tried to write it using the definition, but it didn't work. Please, help me.

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It is not true in general.

Let $\phi(x) = x$, and choose $q=0$, and $q_n = (-1)^n\frac{1}{n}$. Then $\frac{q_n-q}{\|q_n-q\|} = (-1)^n$, and hence $\frac{D\phi(q)(q_n-q)}{\|q_n-q\|} = (-1)^n$, which has no limit as $n \to \infty$.

What is true, however, is that $\lim_n (\frac{\phi(q_n)-\phi(q)}{\|q_n-q\|} - \frac{D\phi(q)(q_n-q)}{\|q_n-q\|}) = 0$.

To see this:

By definition of $D\phi(q)$, you have for all $\epsilon>0$, there exists a $\delta>0$ such that if $\|q'-q\|< \delta$, then $\|\phi(q')-\phi(q) - D\phi(q)(q'-q)\| \leq \epsilon \|q'-q\|$.

Let $\epsilon >0$, then for $n$ sufficiently large we have $\|q_n-q\| < \delta$, and so $\|\phi(q_n)-\phi(q) - D\phi(q)(q_n-q)\| \leq \epsilon \|q_n-q\|$.

If we assume that $q_n \neq q$ for all $n$ sufficiently large, we can divide across to get $\|\frac{\phi(q_n)-\phi(q)}{\|q_n-q\|} - \frac{D\phi(q)(q_n-q)}{\|q_n-q\|}\| \leq \epsilon$, from which it follows that $\lim_n (\frac{\phi(q_n)-\phi(q)}{\|q_n-q\|} - \frac{D\phi(q)(q_n-q)}{\|q_n-q\|}) = 0$.

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But I thought about such explanation: $\lim_{n \to \infty} \frac{D \phi(q)(q_n -q)}{||q_n -q||} = \lim_{n \to \infty} \frac{\phi(q+q_n-q)-\phi(q) - r(||q_n -q||)}{||q_n-q||} = \lim_{n \to \infty} \frac{\phi(q_n) - \phi(q)}{||q_n-q||} - \frac{r(||q_n-q||)}{||q_n-q||}$ and so we get our equation since, by definition, $\lim_{n \to \infty} \frac{r(||q_n-q||)}{||q_n-q||} =0$. So isn't it ok? Why? – Anne Jan 27 '13 at 15:27
Oh, and if we know that $\lim_{n \to \infty} \frac{D\phi(q)(q_n-q)}{||q_n-q||}$ has a limit- is it true then? – Anne Jan 27 '13 at 15:33
Yes, since if $a_n-b_n$ has a limit and $b_n$ has a limit, then $a_n = (a_n-b_n)+b_n$ has a limit. – copper.hat Jan 27 '13 at 17:18

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