I'm looking at a proof that wants to show for $d\geq3$, $y\in\mathbb{R}^d$ and $x\in B_r(0)$ $$\lim_{h\rightarrow 0} \int_{B_r(x)} \frac{y_j}{h}\left(\frac{1}{|y-hu_i|^d}-\frac{1}{|y|^d}\right) (f(y+x)-f(x))dy= \\d\int_{B_r(x)} \frac{y_j y_i}{|y|^{d+2}}(f(y+x)-f(x))dy$$ with $u_i=(0,...,0,1,0,...,0)$, $f$ Hölder-continuous and || as Euclidean norm.
First of all the right-hand side should be finite because of the Hölder cont. The equality should be seen by integrating separately over $|y|\leq h/2$, $h/2<|y|\leq 2h$, $2h<|y|\leq s$ and $y\in B_r(x)\setminus B_s(0)$ and using Hölder cont.
I really can't see how these arguments work. Maybe someone could shine some light on this? I'd appreciate it. (And sorry for the unspecific title.)