I calculated a limit of function as follows:
$$ \begin{array}{ll} \lim_{x \to 1}\frac{x - 1}{x^2 - 1} = \\ \\ \quad = \lim_{x \to 1}\frac{x - 1}{(x + 1)(x - 1)} = \\ \\ \quad = \lim_{x \to 1}\frac{1}{(x + 1)} = \\ \\ \quad = \frac{1}{2} \end{array} $$
Is it valid to minimize the denominator, like I did in the transition from line #2 to line #3 ?
Doesn't it make the function to lose some "properties", therefore making the calculation invalid?
Or maybe it is valid because that we calculate the limit for $x \to 1$ (i.e. a positive number).
If we were to calculate the limit for $x \to (-1)$ (negative number), then the operation wouldn't be valid.
BTW I'm not interested in solving this with L'Hôpital's rule.
This operation is valid. Think of at as considering your fraction defined on $\Bbb R\setminus\{\pm 1\}$, on this set the function is continuous and both numerator and denominator are well-defined and non-zero. The function would not lose any properties if you simplify it.
If we were to calculate the limit $x\to -1$, then whatever we do, the limit does not exist: the numerator converges to $-2$, yet the denominator converges to $0$, therefore the limit of the fraction does not exist.