1. **Problem:** Determine if the function $f(x) = x^4 + 3x^2 - 6x + 2$ is continuous at $x=3$. 2. **Formula and rules:** A polynomial function is continuous everywhere. To check continuity at $x=3$, verify: - $f(3)$ is defined. - $\\lim_{x \to 3} f(x)$ exists. - $f(3) = \\lim_{x \to 3} f(x)$. 3. **Evaluate $f(3)$:** $$f(3) = 3^4 + 3(3^2) - 6(3) + 2 = 81 + 27 - 18 + 2 = 92$$ 4. **Evaluate the limit:** Since $f$ is a polynomial, the limit as $x \to 3$ is simply $f(3)$: $$\\lim_{x \to 3} f(x) = 92$$ 5. **Conclusion:** Since $f(3)$ is defined and equals the limit, $f$ is continuous at $x=3$. Final answer: $f$ is continuous at $x=3$.