1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{x^2 - 6x + 5}{x - 5}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form like $$\frac{0}{0}$$, we try to simplify the expression by factoring or algebraic manipulation. 3. **Evaluate direct substitution:** Substitute $x=5$: $$\frac{5^2 - 6\cdot5 + 5}{5 - 5} = \frac{25 - 30 + 5}{0} = \frac{0}{0}$$ which is indeterminate. 4. **Factor the numerator:** $$x^2 - 6x + 5 = (x - 5)(x - 1)$$ 5. **Rewrite the limit:** $$\lim_{x \to 5} \frac{(x - 5)(x - 1)}{x - 5}$$ 6. **Cancel common factors:** $$\lim_{x \to 5} \frac{\cancel{(x - 5)}(x - 1)}{\cancel{(x - 5)}} = \lim_{x \to 5} (x - 1)$$ 7. **Evaluate the simplified limit:** $$5 - 1 = 4$$ **Final answer:** $$\boxed{4}$$