Just wondering...came across this relationship regarding Euler's number in my math tinkerings, but I'm unaware if this particular relationship has a specific name or not:
$$\lim_{x\to\infty}\frac{x^x}{(x-1)^{x-1}} - \frac{(x-1)^{x-1}}{(x-2)^{x-2}} = e$$
So does it have a formal name? Thanks in advance!
Almost certainly not! There are infinitely many identities like this.
The reason this works is that the two terms are $\left(1-\frac{1}{x}\right)^{-x} \times (x-1)$ and $\left(1-\frac{1}{x-1}\right)^{-(x-1)} \times (x-2)$, where in both cases the first term $\to e$ as $x\to\infty$ by a statement closely related one of the standard definitions of $e$.
Thus this is more a corollary of the statement $\lim_{n\to\infty}(1+1/n)^n = e$ than an independent fact.