1. **Stating the problem:** Calculate the limit $$\lim_{x \to 2} \frac{3x^2 - 5x + 1}{4x - 2}$$. 2. **Formula and rules:** To find limits of rational functions as $x$ approaches a value, first try direct substitution. If it results in a determinate form, that is the limit. If it results in an indeterminate form like $\frac{0}{0}$, factor and simplify. 3. **Calculate numerator at $x=2$:** $$3(2)^2 - 5(2) + 1 = 3 \times 4 - 10 + 1 = 12 - 10 + 1 = 3$$. 4. **Calculate denominator at $x=2$:** $$4(2) - 2 = 8 - 2 = 6$$. 5. **Evaluate limit:** Since denominator is not zero, limit is $$\frac{3}{6} = \frac{1}{2}$$. --- 6. **Stating the problem:** Calculate the limit $$\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4}$$. 7. **Check direct substitution:** Numerator at $x=2$ is $$2^2 - 5 \times 2 + 6 = 4 - 10 + 6 = 0$$. 8. Denominator at $x=2$ is $$2^2 - 4 = 4 - 4 = 0$$. 9. **Indeterminate form $\frac{0}{0}$, so factor numerator and denominator:** Numerator: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ Denominator: $$x^2 - 4 = (x - 2)(x + 2)$$ 10. **Simplify the fraction:** $$\frac{(x - 2)(x - 3)}{(x - 2)(x + 2)} = \frac{x - 3}{x + 2}$$ for $x \neq 2$. 11. **Evaluate the simplified limit at $x=2$:** $$\frac{2 - 3}{2 + 2} = \frac{-1}{4} = -\frac{1}{4}$$. **Final answers:** $$\lim_{x \to 2} \frac{3x^2 - 5x + 1}{4x - 2} = \frac{1}{2}$$ $$\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4} = -\frac{1}{4}$$