Consider Moran's I, a spatial autocorrelation function defined by
$$ I(x) = \frac{N}{W} \frac{\sum\limits_{i,j=1}^{N} w_{ij} (x_i-\langle x\rangle )(x_j-\langle x\rangle ) }{\sum\limits_{i=1}^{N} (x_i-\langle x\rangle )^2 }, $$ where $w_{ij} \geq 0$ are weights, $W = \sum\limits_{i,j=1}^{N} w_{ij} $ is the sum of the weights and $\langle x\rangle = \frac{1}{N} \sum\limits_{i=1}^{N} x_i$ is the mean of $x=(x_1, x_2,\ldots, x_N)$.
This quantity is undefined at $x = 0$, or more generally whenever $x_i = x_j$ for all $i, j$. Suppose we take the metric induced by the norm $|x| = \sum\limits_{i=1}^{N} |x_i|$. I'm interested in the limit $$\lim \limits_{x\rightarrow 0} I(x).$$
What I've tried:
Define $y = x - \langle x \rangle $, then define (functions for the nominator and denominator) $$ f(y) = \sum\limits_{i,j=1}^{N} w_{ij} y_i y_j, \\ g(y) = \sum\limits_{i=1}^{N} y_i^2 $$
If $\forall i$ we have $|y_i| < \delta$, then $$ |f(y)| \leq \sum\limits_{i,j=1}^{N} w_{ij} |y_i| |y_j| < W \delta^2, \\ |g(y)| \leq \sum\limits_{i=1}^{N} |y_i|^2 < N \delta^2 $$ From this we only obtain $|I| < 1$, which is a trivial result.
Another idea was to use l'Hopital's rule for multivariable functions, for which the zero set for $f$ and $g$ would be $C = \{x| x_i = x_j \forall i,j \} = \{y=0\}$. Hence we can take tangent vector $v=(1,0,\ldots, 0)$, so $D_v f(y) = \partial_1 f(y)$ and $D_v g(y) = \partial_1 g(y)$. However, this also does not give the limit and upon reapplying the rule we get $\partial_1^2 f(y) = \partial_1^2 g(y) = 0$.
Notice that $I(tx)=I(x)$ for all $t\in\mathbb{R}\setminus\{0\}$, so in order for $\lim_{x\to0}I(x)$ to exist we would need $I(x)$ to be constant, which is not the case for $N>2$.