Factoring can help us evaluate a limit. For example, we already looked
at the limit
$$\lim_{x \to 1} \frac{x^2-1}{x-1}$$
Since $x^2-1 = (x-1)(x+1)$, we have
$$\lim_{x \to 1} \frac{x^2-1}{x-1} = \lim_{x \to 1}
\frac{(x-1)(x+1)}{x-1} = \lim_{x\to 1} (x+1) = 2.$$
Canceling the factors of $x-1$ from the numerator and denominator is OK
for all values of $x$ except $x=1$. However, we're only looking at
what happens when $x$ is close to 1, not equal to 1, so
the cancelation makes sense.