Consider the expression:
$$\dfrac{(x - c_1) \cdot x}{(x - c_1)}$$
This is often simplified as
$$x \ \text{for} \ x \neq c_1$$
This simplification step can also be done an arbitrary number of times for
$$\dfrac{(x - c_1)(x - c_2) \dots (x - c_n) \cdot x}{(x - c_1)(x - c_2) \dots (x - c_n)}$$
In which case $x \neq c_1, c_2, \dots, c_n$.
Given that it is generally valid to simplify such expressions by repeated steps of elimination of the terms, does that not imply that it is equally valid to introduce arbitrarily many such terms? And if arbitrarily many such terms are introduced, how do we know that $x$ can be defined at all?
I think I can actually explain to myself why this is nonsense.
Given the value x, I think it is intuitive that this could be rewritten as $\dfrac{c1}{c1} \cdot x$ for $c\neq 0$. This is permissible because we haven't arbitrarily introduced an area in a domain where x is undefined. Multiplying by $\dfrac{c}{c} =1$ does not fundamentally change anything. On the other hand, in the case of $\dfrac{x−c}{x−c} \cdot x \ , x \neq c$ , rewriting this as $x$ is only permissible in the case that the restrictions on the domain are maintained. I.e, because the area in which $x$ is not defined is the same before and after the simplification step. Introducing such a term arbitrarily however breaks this rule, and is thus invalid.