A limit describes where the outputs are heading, even when the function is missing at the destination.
For x ≠ 1, factor x² − 1 = (x − 1)(x + 1), so the quotient equals x + 1. At x = 1 the original quotient is undefined. Nearby values still approach 2 from both sides.
Cancelling a factor does not fill the original function’s hole.
The hollow point stays hollow. The original expression is undefined at x = 1.