I came across the following claim
$$\lim_{r \to 0} \dfrac{f(rx) - f(0)}{r} = \langle v,x \rangle$$
locally uniformly implies that the derivative of $f$ at $0$ exists and $Df(0) = v$.
I don't follow how this holds, the definition of the derivative at $a$ is an operator $Df(a)$ satisfying
$$\lim_{r\to a} \dfrac{|f(r)-f(a)-Df(a)|r-a||}{|a-r|} = 0$$
on the other hand,
$$\langle v,x \rangle = |v| |x| \cos \theta$$
What am I missing?