Removing Discontinuities AB & BC

A discontinuity can be removed exactly when the two-sided limit exists, which is what makes it removable. The repair is to define a new function agreeing with the original away from the point and taking the value $\lim_{x\to a} f(x)$ at it. Jump and infinite discontinuities admit no repair, because there is no single limit value to assign, so classifying the break is what determines whether a repair is available.

Two errors dominate. Cancelling a common factor gets mistaken for the repair itself, when the simplification is only a valid identity away from the point and the original function is still undefined there; the cancellation reveals the limit rather than fixing anything. And the repair value gets chosen wrongly, usually by trying to substitute into the original expression, which returns $\frac{0}{0}$. Related to both, factoring gets applied to a genuine asymptote in the hope of removing it, when no manipulation can make an unbounded discontinuity repairable.