Let $z,w\in\mathbb B^n\subset\mathbb C^n$. How can I calculate the following limit? $$\lim\limits_{w\to z}\frac{\langle z-w, a\rangle}{|w-P_w(z)-s_wQ_w(z)|}$$ where $P_w(z)=\frac{\langle z,w\rangle}{|w|^2}w$, $Q_w(z)=z-P_w(z)$ and $s_w=\sqrt{1-|w|^2}$.
If $w=rz$ ($r\in(0,1)$), then it is easy to get that $$\lim\limits_{w\to z}\frac{\langle z-w, a\rangle}{|w-P_w(z)-s_wQ_w(z)|}=\left\langle\frac{z}{|z|},a\right\rangle.$$ I suspect that the limit is the same in general. Can anyone help me to calculate it? Thanks a lot!
The limit should not exist. The tangential limit is $$\left\langle \mathrm i \frac z{|z|},a\right\rangle.$$