1. **State the problem:** Find the derivative of the function $f(x) = 3x^2 - 11$ at $x = 2$ using the limit definition of the derivative. 2. **Recall the limit definition of the derivative:** $$m(2) = \lim_{\Delta x \to 0} \frac{f(2 + \Delta x) - f(2)}{\Delta x}$$ This formula calculates the slope of the tangent line to the curve at $x=2$. 3. **Calculate $f(2)$:** $$f(2) = 3(2)^2 - 11 = 3 \times 4 - 11 = 12 - 11 = 1$$ 4. **Calculate $f(2 + \Delta x)$:** $$f(2 + \Delta x) = 3(2 + \Delta x)^2 - 11 = 3(4 + 4\Delta x + \Delta x^2) - 11 = 12 + 12\Delta x + 3\Delta x^2 - 11 = 1 + 12\Delta x + 3\Delta x^2$$ 5. **Substitute into the limit expression:** $$\frac{f(2 + \Delta x) - f(2)}{\Delta x} = \frac{(1 + 12\Delta x + 3\Delta x^2) - 1}{\Delta x} = \frac{12\Delta x + 3\Delta x^2}{\Delta x}$$ 6. **Simplify the fraction by canceling $\Delta x$:** $$\frac{\cancel{\Delta x}(12 + 3\Delta x)}{\cancel{\Delta x}} = 12 + 3\Delta x$$ 7. **Take the limit as $\Delta x \to 0$:** $$m(2) = \lim_{\Delta x \to 0} (12 + 3\Delta x) = 12 + 3 \times 0 = 12$$ **Final answer:** The derivative of $f(x)$ at $x=2$ is $\boxed{12}$.