1. **State the problem:** Find the limit as $x$ approaches 1 of the expression $$\frac{x^2 - 2x + 1}{x - 1}.$$\n\n2. **Recognize the form:** Substitute $x=1$ directly: numerator $=1^2 - 2(1) + 1 = 1 - 2 + 1 = 0$, denominator $=1 - 1 = 0$. This is an indeterminate form $\frac{0}{0}$, so we need to simplify.\n\n3. **Factor the numerator:** Notice that $x^2 - 2x + 1$ is a perfect square trinomial: $$x^2 - 2x + 1 = (x - 1)^2.$$\n\n4. **Rewrite the expression:** $$\frac{(x - 1)^2}{x - 1}.$$\n\n5. **Simplify by canceling common factors:** $$\frac{\cancel{(x - 1)}(x - 1)}{\cancel{(x - 1)}} = x - 1.$$\n\n6. **Evaluate the limit of the simplified expression:** $$\lim_{x \to 1} (x - 1) = 1 - 1 = 0.$$\n\n**Final answer:** $$\boxed{0}.$$