Is the image of $Df(\bar{x})$ equal to the set of all limit directions of $(f(x)-f(\bar{x}))/\|f(x)-f(\bar{x})\|$ as $x\to \bar{x}$? Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be a continuously differentiable function and let $\bar{x}\in \mathbb{R}^n$. What I call "the set of all limit directions of $f(x)$ as $x\to\bar{x}$"  is the following set: $$L(\bar{x})\doteq \text{span}\{d\in \mathbb{R}^m\mid \exists \{x_k\}_{k\in \mathbb{N}}\to \bar{x}, \text{ such that } \lim_{k\to\infty}\frac{f(x_k)-f(\bar{x})}{\|f(x_k)-f(\bar{x})\|}= d\}.$$ 
It looks like a derivative and it also looks like a linearized version of the image of $f$ around $\bar{x}$. The resemblance is more striking if we assume $f(\bar{x})=0$. Then, my question is:

Is this true: $L(\bar{x})=\text{span}(Df(\bar{x}))$? If not, is any of the inclusions true?

I tried a few examples and I got equality in all of them. Thanks in advance!
 A: I am going to try.
If $Ker(Df(\bar x))=\{0\}$ (which implies, in particular, $m\ge n$ and a local diffemorphism), you can do
$$
f(x_k)-f(\bar x)=Df(\bar x)(x_k-\bar x)+o(x_k-\bar x)
$$
Now you divide both sides by $\|f(x_k)-f(\bar x)\|$ and writing
$$
x_k-\bar x=Df^{-1}(f(\bar x))(f(x_k)-f(\bar x))+o(f(x_k)-f(\bar x))
$$
you get
$$
\frac{f(x_k)-f(\bar x)}{\|f(x_k)-f(\bar x)\|}=Df(\bar x)Df^{-1}(f(\bar x))(\frac{f(x_k)-f(\bar x)}{\|f(x_k)-f(\bar x)\|})+\frac{o(f(x_k)-f(\bar x))}{\|f(x_k)-f(\bar x)\|}+\frac{o(x_k-\bar x)}{\|f(x_k)-f(\bar x)\|}
$$
In the limit you get $d\in L(\bar x)$ on the left hand side. The quotients involving little o's converge to zero, the second one because you rearrange as
$$
\frac{o(x_k-\bar x)}{\|f(x_k)-f(\bar x)\|}=\frac{o(x_k-\bar x)}{\|x_k-\bar x\|}\frac{\|x_k-\bar x\|}{\|f(x_k)-f(\bar x)\|}
$$
which converges to zero because the second factor is bounded by our assumption on invertibility.
You conclude that 
$$
d=Df(\bar x)(w)
$$
where $w=\lim Df^{-1}(f(\bar x))(f(x_k)-f(\bar x))$.
If the condition does not hold, you can take quotients by the kernel of $Df(\bar x)$ and the same proof works, replacing norms by norms in the quotient and the linear operators by their induced linear operators in the quotient, etc.
