1. **State the problem:** We need to find the derivative of the function $f(x) = 3x^2 - 7x + 1$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ and then evaluate $f'(x)$ at $x=2$. 2. **Apply the limit definition:** Substitute $f(x+h)$ and $f(x)$: $$f(x+h) = 3(x+h)^2 - 7(x+h) + 1$$ $$= 3(x^2 + 2xh + h^2) - 7x - 7h + 1$$ $$= 3x^2 + 6xh + 3h^2 - 7x - 7h + 1$$ 3. **Calculate the difference quotient:** $$\frac{f(x+h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2 - 7x - 7h + 1) - (3x^2 - 7x + 1)}{h}$$ Simplify the numerator: $$= \frac{3x^2 + 6xh + 3h^2 - 7x - 7h + 1 - 3x^2 + 7x - 1}{h}$$ $$= \frac{6xh + 3h^2 - 7h}{h}$$ 4. **Cancel common factor $h$:** $$= \frac{\cancel{h}(6x + 3h - 7)}{\cancel{h}} = 6x + 3h - 7$$ 5. **Take the limit as $h \to 0$:** $$f'(x) = \lim_{h \to 0} (6x + 3h - 7) = 6x - 7$$ 6. **Evaluate at $x=2$:** $$f'(2) = 6(2) - 7 = 12 - 7 = 5$$ **Final answer:** $$f'(x) = 6x - 7$$ $$f'(2) = 5$$