1. The problem is to understand the sum and difference rules for limits in calculus. 2. The sum rule states that the limit of a sum is the sum of the limits: $$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$ provided both limits exist. 3. The difference rule states that the limit of a difference is the difference of the limits: $$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$ provided both limits exist. 4. These rules allow us to break complex limits into simpler parts. 5. Example: Find $$\lim_{x \to 2} (3x + 5 - (x^2 - 1))$$. 6. Using the difference rule: $$\lim_{x \to 2} (3x + 5) - \lim_{x \to 2} (x^2 - 1)$$. 7. Calculate each limit separately: $$3(2) + 5 = 6 + 5 = 11$$ and $$2^2 - 1 = 4 - 1 = 3$$. 8. Subtract the results: $$11 - 3 = 8$$. 9. Therefore, $$\lim_{x \to 2} (3x + 5 - (x^2 - 1)) = 8$$. 10. Practice problems: (1) $$\lim_{x \to 1} (x + 4 + 2x)$$ (2) $$\lim_{x \to 3} (5x - 7 - x^2)$$ (3) $$\lim_{x \to 0} (x^2 + 3x)$$ (4) $$\lim_{x \to -1} (2x - 3 + x^3)$$ (5) $$\lim_{x \to 2} (4x - 5 - 2x)$$ (6) $$\lim_{x \to 5} (x^2 + x - 10)$$ (7) $$\lim_{x \to 0} (7x - 4x^2)$$ (8) $$\lim_{x \to 1} (3x + 2 - x)$$ (9) $$\lim_{x \to 4} (x^2 - 2x + 1)$$ (10) $$\lim_{x \to 3} (6x - 9 - 3x)$$ These problems use sum and difference rules to find limits easily.