I have a quick question about the proper way to take the coalescent limit for a divided difference formula. It seems to me that there are two ways that we could conceivably define the coalescent limit, and I am having trouble finding any resources that definitively say which way is correct (most of them say that you just "take the coalescent limit" or "take the limit as points coalesce" but give no description for the procedure).
The main distinction here will be the difference between taking the coalescent limit of points pairwise versus taking the limit all at once. Suppose that we want to take the coalescent limit of a second order divided difference on general points $x_0,x_1,x_2$. If we were to do this sequentially, then we might define the coalescent limit to the point $x_0$ as follows
$$\begin{align*} \underset{x_1\rightarrow x_0}{\text{c-lim}}\,u[x_0,x_1] &:= \lim_{x_1\rightarrow x_0} \frac{u \! \left(x_{0}\right)-u \! \left(x_{1}\right)}{x_{0}-x_{1}} = u'(x_0)\\ \underset{x_1,x_2\rightarrow x_0}{\text{c-lim}}\,u[x_0,x_1,x_2] &:=\lim_{x_2\rightarrow x_0}\lim_{x_1\rightarrow x_0}\frac{\left(x_{2}-x_{0}\right) u \! \left(x_{1}\right)+\left(x_{0}-x_{1}\right) u \! \left(x_{2}\right)+u \! \left(x_{0}\right) \left(x_{1}-x_{2}\right)}{\left(x_{0}-x_{1}\right) \left(x_{0}-x_{2}\right) \left(x_{1}-x_{2}\right)}\\ &\qquad=\lim_{x_1\rightarrow x_0}\frac{u'\left(x_{0}\right) x_{0}-u' \left(x_{0}\right) x_{1}-u \! \left(x_{0}\right)+u \! \left(x_{1}\right)}{x_{0}^{2}-2 x_{0} x_{1}+x_{1}^{2}}\\ &\qquad=\frac{u''(x_0)}{2}. \end{align*}$$
Of course, in the evaluation of $\underset{x_1,x_2\rightarrow x_0}{\text{c-lim}}\,u[x_0,x_1,x_2]$ we could've interchanged the order of the limit, and the answer would've been unaffected. However, if we define the limit for simultaneous coalescence (that is to say, we force all the points to reach $x_0$ at the same time), we can get
$$\begin{align*} \underset{x_1,x_2\rightarrow x_0}{\text{c-lim}}\,u[x_0,x_1,x_2] &:=\lim_{t\rightarrow 0} \frac{[(x_{2}-x_{0}) t]\, u (x_{0}+(x_{1}-x_{0}) t )-[(x_{1}-x_{0}) t]\, u (x_{0}+(x_{2}-x_{0}) t )+u (x_{0}) [(x_{1}-x_{0}) t -(x_{2}-x_{0}) t ]}{(x_{1}-x_{0}) t^{2} \left(x_{2}-x_{0}\right) [\left(x_{1}-x_{0}\right) t -\left(x_{2}-x_{0}\right) t ]} \\ &\qquad=\frac{u''(x_0)}{2}. \end{align*}$$
So my question is which one of these is correct? I am mainly interested in this since, depending on the definition that you take, it either does or does not make sense to take the coalescent limit of an expression of the form $$\frac{x_1-x_0}{x_2-x_0}.$$
Any help is appreciated, and if anyone can point me in a direction of a resource that deals with this issue specifically (again, all the numerical analysis references that I can find just tell you to take the limit without being explicit about how to do this), that would be phenomenal. Thank you!