From Förberedande kurs i matematik 2

The numerator can be factorized using the formula for the difference of two squares to give \displaystyle x^2-1=(x+1)(x-1) and then we see that the numerator and denominator have a common factor which we can eliminate

\displaystyle \frac{x^{2}-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1\,\textrm{.}