1. **State the problem:** Find the limit $$\lim_{x \to 4} \frac{3x^2 - 14x + 8}{x^2 - 3x - 4}$$. 2. **Check direct substitution:** Substitute $x=4$ into numerator and denominator. Numerator: $$3(4)^2 - 14(4) + 8 = 3 \times 16 - 56 + 8 = 48 - 56 + 8 = 0$$ Denominator: $$4^2 - 3(4) - 4 = 16 - 12 - 4 = 0$$ Since direct substitution gives $\frac{0}{0}$, an indeterminate form, we need to simplify. 3. **Factor numerator and denominator:** Numerator: $$3x^2 - 14x + 8$$ Find factors of $3 \times 8 = 24$ that sum to $-14$: $-12$ and $-2$. Rewrite numerator: $$3x^2 - 12x - 2x + 8 = 3x(x - 4) - 2(x - 4) = (3x - 2)(x - 4)$$ Denominator: $$x^2 - 3x - 4$$ Factor: $$(x - 4)(x + 1)$$ 4. **Simplify the fraction:** $$\frac{(3x - 2)(x - 4)}{(x - 4)(x + 1)}$$ Cancel common factor $(x - 4)$: $$\frac{\cancel{(x - 4)}(3x - 2)}{\cancel{(x - 4)}(x + 1)} = \frac{3x - 2}{x + 1}$$ 5. **Evaluate the limit of simplified expression:** Substitute $x=4$: $$\frac{3(4) - 2}{4 + 1} = \frac{12 - 2}{5} = \frac{10}{5} = 2$$ **Final answer:** $$\boxed{2}$$