1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{x^2 - 2x + 1}{|x - 1|}$$. 2. **Rewrite the expression:** Notice that the numerator can be factored as $$x^2 - 2x + 1 = (x - 1)^2$$. 3. **Substitute the factorization:** The limit becomes $$\lim_{x \to 1} \frac{(x - 1)^2}{|x - 1|}$$. 4. **Simplify the expression:** Since $$|x - 1|$$ is the absolute value of $$x - 1$$, we can write: $$\frac{(x - 1)^2}{|x - 1|} = \frac{|x - 1|^2}{|x - 1|} = |x - 1|$$. 5. **Evaluate the limit:** As $$x \to 1$$, $$|x - 1| \to 0$$. Therefore, $$\lim_{x \to 1} \frac{x^2 - 2x + 1}{|x - 1|} = 0$$. **Final answer:** $$0$$