1. **State the problem:** Find the limits: $$\lim_{x \to 5} (3x^2 - 1)$$ and $$\lim_{x \to -2} (3x^3 + x^2 - 1)$$ 2. **Recall the limit theorem for polynomials:** For any polynomial function $f(x)$, the limit as $x$ approaches a value $a$ is simply $f(a)$ because polynomials are continuous everywhere. 3. **Evaluate the first limit:** Substitute $x=5$ into $3x^2 - 1$: $$3(5)^2 - 1 = 3 \times 25 - 1 = 75 - 1 = 74$$ 4. **Evaluate the second limit:** Substitute $x=-2$ into $3x^3 + x^2 - 1$: $$3(-2)^3 + (-2)^2 - 1 = 3(-8) + 4 - 1 = -24 + 4 - 1 = -21$$ 5. **Summary:** $$\lim_{x \to 5} (3x^2 - 1) = 74$$ $$\lim_{x \to -2} (3x^3 + x^2 - 1) = -21$$ These results are confirmed by the continuity of polynomial functions and can be verified by a table of values approaching the limit points.