It works because you are evaluating the limit at that point. Think of it as multiplying both sides by (x-1) and then applying the limit as x->1 to both sides.

It works because you are evaluating the limit at that point. Think of it as multiplying both sides by (x-1) and then applying the limit as x->1 to both sides. When you do that, you are going to end with an equality, 1 = C.