What would be the ideal formal notation to express that, as a variable $X$ becomes more similar to number $a$, the function $f(X)$ always becomes more similar to number $z$?
As far as I understand, both the following two formal constructs are actually inaccurate:
- as $X \to a$, $f(X) \to z$
- $\lim\limits_{X \to a} f(X)=z$
The reason why I think they are both not accurate is that they guarantee only that when $X$ is arbitrarily close to $a$, $f(X)$ becomes arbitrarily close to $z$. Yet, they do not imply a monotonic relation, that is, they do not imply that as $X$ approximates $a$, $f(X)$ always gets closer to $z$.
If I understand correctly, then I propose something like "For all $x_1,x_2$: $|x_2-a|<|x_1-a|\implies |f(x_2)-z|<|f(x_1)-z|$" If you also want the difference between $f$ and $z$ to get arbitrarily small, you could add "and $\displaystyle{\lim_{x\to a}}f(x)=z$".