From Förberedande kurs i matematik 2

The numerator can be factorized using the conjugate rule to give \displaystyle x^{2}-1=\left( x+1 \right)\left( x-1 \right) 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{\left( x+1 \right)\left( x-1 \right)}{x-1}=x+1