1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{(5 - 2x)^2 - 243}{x - 1}$$. 2. **Recall the formula and approach:** This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ as $$x \to a$$, which resembles the definition of the derivative of $$f(x)$$ at $$x = a$$. 3. **Identify the function:** Let $$f(x) = (5 - 2x)^2$$ and $$a = 1$$. 4. **Calculate $$f(a)$$:** $$f(1) = (5 - 2 \cdot 1)^2 = (5 - 2)^2 = 3^2 = 9$$. 5. **Rewrite the numerator:** $$ (5 - 2x)^2 - 243 = (5 - 2x)^2 - 9^2 $$ 6. **Factor the difference of squares:** $$ (5 - 2x)^2 - 9^2 = \left((5 - 2x) - 9\right) \left((5 - 2x) + 9\right) = (5 - 2x - 9)(5 - 2x + 9) $$ 7. **Simplify factors:** $$ (5 - 2x - 9) = -4 - 2x $$ $$ (5 - 2x + 9) = 14 - 2x $$ 8. **Rewrite the limit:** $$ \lim_{x \to 1} \frac{(-4 - 2x)(14 - 2x)}{x - 1} $$ 9. **Notice direct substitution:** Substituting $$x=1$$ gives numerator $$(-4 - 2)(14 - 2) = (-6)(12) = -72$$ and denominator $$0$$, so direct substitution is undefined. 10. **Use derivative approach:** The limit is the derivative of $$f(x)$$ at $$x=1$$: $$ f'(x) = 2(5 - 2x)(-2) = -4(5 - 2x) $$ 11. **Evaluate derivative at $$x=1$$:** $$ f'(1) = -4(5 - 2 \cdot 1) = -4(5 - 2) = -4 \cdot 3 = -12 $$ 12. **Final answer:** $$ \lim_{x \to 1} \frac{(5 - 2x)^2 - 243}{x - 1} = -12 $$