1. Problem 49: Given $$\lim_{x \to 4} \frac{f(x) - 5}{x - 2} = 1$$, find $$\lim_{x \to 4} f(x)$$. 2. We use the property of limits and continuity. The expression resembles the definition of a derivative but here we focus on the limit of $$f(x)$$. 3. Substitute $$x = 4$$ in the denominator: $$4 - 2 = 2$$. 4. Rewrite the limit as $$\lim_{x \to 4} \frac{f(x) - 5}{2} = 1$$. 5. Multiply both sides by 2: $$\lim_{x \to 4} (f(x) - 5) = 2$$. 6. Add 5 to both sides: $$\lim_{x \to 4} f(x) = 7$$. --- 7. Problem 50: Given $$\lim_{x \to -2} \frac{f(x)}{x^2} = 1$$. (a) Find $$\lim_{x \to -2} f(x)$$. (b) Find $$\lim_{x \to -2} \frac{f(x)}{x}$$. 8. For (a), since $$\lim_{x \to -2} \frac{f(x)}{x^2} = 1$$ and $$x^2$$ is continuous, $$\lim_{x \to -2} x^2 = (-2)^2 = 4$$. 9. Multiply both sides by 4: $$\lim_{x \to -2} f(x) = 1 \times 4 = 4$$. 10. For (b), $$\lim_{x \to -2} \frac{f(x)}{x} = \lim_{x \to -2} f(x) \times \lim_{x \to -2} \frac{1}{x} = 4 \times \frac{1}{-2} = -2$$. --- 11. Problem 51: (a) Given $$\lim_{x \to 2} \frac{f(x) - 5}{x - 2} = 3$$, find $$\lim_{x \to 2} f(x)$$. (b) Given $$\lim_{x \to 2} \frac{f(x) - 5}{x - 2} = 4$$, find $$\lim_{x \to 2} f(x)$$. 12. Similar to problem 49, substitute $$x = 2$$ in denominator: $$2 - 2 = 0$$, but the limit exists, so numerator must approach 0. 13. Rewrite limit as $$\lim_{x \to 2} \frac{f(x) - 5}{x - 2} = L$$, where $$L$$ is 3 or 4. 14. Multiply both sides by $$x - 2$$ and take limit: $$\lim_{x \to 2} (f(x) - 5) = L \times 0 = 0$$. 15. So, $$\lim_{x \to 2} f(x) = 5$$ for both (a) and (b). --- 16. Problem 52: Given $$\lim_{x \to 0} \frac{f(x)}{x^2} = 1$$. (a) Find $$\lim_{x \to 0} f(x)$$. (b) Find $$\lim_{x \to 0} \frac{f(x)}{x}$$. 17. For (a), since $$\lim_{x \to 0} \frac{f(x)}{x^2} = 1$$ and $$\lim_{x \to 0} x^2 = 0$$, multiply both sides by 0: $$\lim_{x \to 0} f(x) = 1 \times 0 = 0$$. 18. For (b), $$\lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} f(x) \times \lim_{x \to 0} \frac{1}{x}$$. 19. But $$\lim_{x \to 0} \frac{1}{x}$$ does not exist (infinite), so the limit does not exist. Final answers: 49. $$\lim_{x \to 4} f(x) = 7$$. 50. (a) $$4$$, (b) $$-2$$. 51. (a) $$5$$, (b) $$5$$. 52. (a) $$0$$, (b) does not exist.