1. **State the problem:** Evaluate the limit $$\lim_{x \to 2} \frac{3x^3 - 6x^2}{x^2 - 2x}.$$\n\n2. **Substitute $x=2$ to check if direct evaluation works:**\nCalculate numerator: $3(2)^3 - 6(2)^2 = 3 \times 8 - 6 \times 4 = 24 - 24 = 0$.\nCalculate denominator: $(2)^2 - 2 \times 2 = 4 - 4 = 0$.\nDirect substitution results in the indeterminate form $\frac{0}{0}$, so we must simplify.\n\n3. **Factor numerator and denominator:**\nNumerator: $3x^3 - 6x^2 = 3x^2(x - 2)$.\nDenominator: $x^2 - 2x = x(x - 2)$.\n\n4. **Rewrite the expression:**\n$$\frac{3x^3 - 6x^2}{x^2 - 2x} = \frac{3x^2(x - 2)}{x(x - 2)}.$$\n\n5. **Cancel the common factor $(x - 2)$:**\n$$\frac{3x^2 \cancel{(x - 2)}}{x \cancel{(x - 2)}} = \frac{3x^2}{x} = 3x.$$\n\n6. **Evaluate the limit of the simplified expression:**\n$$\lim_{x \to 2} 3x = 3 \times 2 = 6.$$\n\n**Final answer:** $$6.$$