Just a references and a summary to think about:
Instead of the OP's $(x^2-1)/(x-1)$ another function is considered, namely the Sinc function,
which has important physical applications in for example spectrography and signal processing:
$$
\operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\
1 & \text{for } x =0 \end{cases}
$$
It is argued in the reference that nobody in physics would ever think of a definition with a discontinuity at $x=0$ such as:
$$
\operatorname{suck}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\
0 & \text{for } x =0 \end{cases}
$$
The basic reason is that, from a physics viewpoint, continuity is very similar to equality.
In order to appreciate this, you might have to notice an equivalence between the $(\epsilon,\delta)$ formalism in analysis and the processing of errors in physics.
The continuum in physics is always error prone. There is no other equality there than approximately equal. That's why we have the following sequence of abstractions:
$$
|x-a| \lt \delta \quad \Longrightarrow \quad |f(x)-f(a)| \lt \epsilon \\
x \approx a \quad \Longrightarrow \quad f(x) \approx f(a) \\
x=a \quad \Longrightarrow \quad f(x)=f(a)
$$
With other words: functions defined on a continuum in the real world can be only functions if they are themselves continuous as well.
All this in a nutshell. More details are in the reference as mentioned. Alas, there is much more to tell that doesn't fit into the margins of MSE :-(
As far as the OP's function is concerned, indeed: $(x^2-1)/(x-1)=x+1$ is always true in the real world. No black holes are going to change that fact.