1. **State the problem:** Find the limit as $x \to 3$ of the function $f(x) = -2x^2 + 7x - 3$. 2. **Recall the limit rule for polynomials:** The limit of a polynomial function as $x$ approaches a value $a$ is simply the value of the polynomial evaluated at $a$ because polynomials are continuous everywhere. 3. **Evaluate the polynomial at $x=3$:** $$f(3) = -2(3)^2 + 7(3) - 3$$ 4. **Calculate step-by-step:** $$= -2 \times 9 + 21 - 3$$ $$= -18 + 21 - 3$$ 5. **Simplify:** $$= ( -18 + 21 ) - 3 = 3 - 3 = 0$$ 6. **Conclusion:** The limit is $$\boxed{0}$$ This means as $x$ approaches 3, the function value approaches 0.