Minimizing denominator when calculating limit of function

Question

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.

Answer 1

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.