1. **Problem Statement:** Determine the following limits: a) $$\lim_{x \to +\infty} (3x^4 - 5x^2 + 8)$$ b) $$\lim_{x \to -\infty} (4x^3 + 10x^2 + 11)$$ c) $$\lim_{x \to -\infty} (-x^2 + 3x + 4)$$ d) $$\lim_{x \to +\infty} (1 + 3x + 5x^2 - x^3)$$ 2. **Formula and Rules:** When finding limits at infinity for polynomials, the term with the highest power dominates the behavior. - If the highest power term has a positive coefficient and the power is even, the limit as $$x \to \pm\infty$$ is $$+\infty$$. - If the highest power term has a positive coefficient and the power is odd, the limit as $$x \to +\infty$$ is $$+\infty$$ and as $$x \to -\infty$$ is $$-\infty$$. - If the highest power term has a negative coefficient, the limits reverse accordingly. 3. **Step-by-step Solutions:** a) $$3x^4 - 5x^2 + 8$$ as $$x \to +\infty$$ - Highest power term: $$3x^4$$ (power 4, even, coefficient positive) - As $$x \to +\infty$$, $$3x^4 \to +\infty$$ dominates. - Therefore, $$\lim_{x \to +\infty} (3x^4 - 5x^2 + 8) = +\infty$$. b) $$4x^3 + 10x^2 + 11$$ as $$x \to -\infty$$ - Highest power term: $$4x^3$$ (power 3, odd, coefficient positive) - As $$x \to -\infty$$, $$4x^3 \to -\infty$$ dominates. - Therefore, $$\lim_{x \to -\infty} (4x^3 + 10x^2 + 11) = -\infty$$. c) $$-x^2 + 3x + 4$$ as $$x \to -\infty$$ - Highest power term: $$-x^2$$ (power 2, even, coefficient negative) - As $$x \to -\infty$$, $$-x^2 \to -\infty$$ dominates. - Therefore, $$\lim_{x \to -\infty} (-x^2 + 3x + 4) = -\infty$$. d) $$1 + 3x + 5x^2 - x^3$$ as $$x \to +\infty$$ - Highest power term: $$-x^3$$ (power 3, odd, coefficient negative) - As $$x \to +\infty$$, $$-x^3 \to -\infty$$ dominates. - Therefore, $$\lim_{x \to +\infty} (1 + 3x + 5x^2 - x^3) = -\infty$$. 4. **Final answers:** a) $$+\infty$$ b) $$-\infty$$ c) $$-\infty$$ d) $$-\infty$$