1. **State the problem:** We need to evaluate the limit $$\lim_{x \to 6^-} \frac{x^2 - 4x + 3}{x - 1}$$ which means finding the value the expression approaches as $x$ approaches 6 from the left. 2. **Recall the formula and rules:** The limit of a rational function as $x$ approaches a point can be found by direct substitution if the function is defined at that point. If direct substitution leads to an indeterminate form, we simplify the expression. 3. **Simplify the numerator:** Factor the quadratic expression: $$x^2 - 4x + 3 = (x - 3)(x - 1)$$ 4. **Rewrite the expression:** $$\frac{x^2 - 4x + 3}{x - 1} = \frac{(x - 3)(x - 1)}{x - 1}$$ 5. **Cancel common factors:** Since $x \neq 1$ (we are approaching 6), we can cancel $x - 1$: $$= x - 3$$ 6. **Evaluate the limit:** Substitute $x = 6$: $$6 - 3 = 3$$ 7. **Conclusion:** The limit is 3. **Final answer:** 3