1. Problem 5: Find $$\lim_{y \to 2^-} \frac{(y-1)(y-2)}{y+1}$$. Formula: For limits involving rational functions, if direct substitution leads to a defined value, that is the limit. If it leads to an indeterminate form, factor and simplify. Work: Direct substitution: $$\frac{(2-1)(2-2)}{2+1} = \frac{1 \cdot 0}{3} = 0$$. Since the numerator approaches 0 and denominator approaches 3, limit is 0. 2. Problem 6: $$\lim_{x \to 3} \frac{x^2 - 2x}{x+1}$$. Direct substitution: $$\frac{9 - 6}{4} = \frac{3}{4}$$. Limit is $$\frac{3}{4}$$. 3. Problem 7: $$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$$. Direct substitution: denominator 0, numerator 0, indeterminate. Factor numerator: $$x^2 - 16 = (x-4)(x+4)$$. Simplify: $$\frac{(x-4)(x+4)}{x-4} = x+4$$ for $$x \neq 4$$. Limit: $$4 + 4 = 8$$. 4. Problem 8: $$\lim_{x \to 0^+} \frac{6x - 9}{x^2 - 12x + 3}$$. Substitute 0: numerator $$-9$$, denominator $$3$$. Limit: $$\frac{-9}{3} = -3$$. 5. Problem 9: $$\lim_{x \to 1^+} \frac{x^4 - 1}{x - 1}$$. Indeterminate form 0/0. Factor numerator: $$x^4 - 1 = (x-1)(x^3 + x^2 + x + 1)$$. Simplify: $$x^3 + x^2 + x + 1$$. Evaluate at 1: $$1 + 1 + 1 + 1 = 4$$. 6. Problem 10: $$\lim_{x \to 2} \frac{x^3 + 8}{1 + 2} = \frac{8 + 8}{3} = \frac{16}{3}$$. 7. Problem 11: $$\lim_{x \to -1} \frac{x^2 + 6x + 5}{x^2 - 3x - 4}$$. Factor numerator: $$(x+5)(x+1)$$. Factor denominator: $$(x-4)(x+1)$$. Simplify: $$\frac{x+5}{x-4}$$ for $$x \neq -1$$. Evaluate at $$-1$$: $$\frac{-1+5}{-1-4} = \frac{4}{-5} = -\frac{4}{5}$$. 8. Problem 12: $$\lim_{x \to 2} \frac{x^2 - 4x + 4}{x^2 + x - 6}$$. Numerator: $$(x-2)^2$$. Denominator: $$(x+3)(x-2)$$. Simplify: $$\frac{x-2}{x+3}$$ for $$x \neq 2$$. Evaluate at 2: $$\frac{0}{5} = 0$$. 9. Problem 13: $$\lim_{x \to 2} \frac{x^3 + 3x^2 - 12x + 4}{x^3 - 4x}$$. Evaluate numerator at 2: $$8 + 12 - 24 + 4 = 0$$. Evaluate denominator at 2: $$8 - 8 = 0$$. Indeterminate form. Factor numerator: Try synthetic division by (x-2). Divide numerator by (x-2): quotient $$x^2 + 5x - 2$$. Denominator: $$x(x^2 - 4) = x(x-2)(x+2)$$. Simplify limit expression: $$\frac{(x-2)(x^2 + 5x - 2)}{x(x-2)(x+2)} = \frac{x^2 + 5x - 2}{x(x+2)}$$ for $$x \neq 2$$. Evaluate at 2: Numerator: $$4 + 10 - 2 = 12$$. Denominator: $$2 \cdot 4 = 8$$. Limit: $$\frac{12}{8} = \frac{3}{2}$$. 10. Problem 14: $$\lim_{x \to 1} \frac{x^3 + x^2 - 5x + 3}{x^3 - 3x + 2}$$. Evaluate numerator at 1: $$1 + 1 - 5 + 3 = 0$$. Evaluate denominator at 1: $$1 - 3 + 2 = 0$$. Indeterminate. Factor numerator by synthetic division with (x-1): quotient $$x^2 + 2x - 3$$. Factor quotient: $$(x+3)(x-1)$$. Numerator: $$(x-1)(x+3)(x-1) = (x-1)^2 (x+3)$$. Denominator: factor $$x^3 - 3x + 2$$. Try (x-1): synthetic division quotient $$x^2 + x - 2$$. Factor quotient: $$(x+2)(x-1)$$. Denominator: $$(x-1)^2 (x+2)$$. Simplify: $$\frac{(x-1)^2 (x+3)}{(x-1)^2 (x+2)} = \frac{x+3}{x+2}$$ for $$x \neq 1$$. Evaluate at 1: $$\frac{4}{3}$$. 11. Problem 15: $$\lim_{x \to 3^-} \frac{x}{x-3}$$. As $$x \to 3^-$$, denominator $$x-3 \to 0^-$$. Numerator $$\to 3$$. Limit: $$\frac{3}{0^-} = -\infty$$. 12. Problem 16: $$\lim_{x \to 3} \frac{x}{x-3}$$. Left limit: $$-\infty$$, right limit: $$+\infty$$. Limit does not exist. 13. Problem 17: $$\lim_{x \to 3^+} \frac{x}{x-3}$$. Denominator $$0^+$$, numerator $$3$$. Limit: $$+\infty$$. 14. Problem 18: $$\lim_{x \to 2^-} \frac{x}{x^2 - 4}$$. Denominator $$x^2 - 4 = (x-2)(x+2)$$. At $$x=2$$ denominator 0. For $$x \to 2^-$$, $$x-2 \to 0^-$$, $$x+2 \to 4$$. Denominator $$\to 0^-$$. Numerator $$\to 2$$. Limit: $$\frac{2}{0^-} = -\infty$$. 15. Problem 19: $$\lim_{x \to 2^+} \frac{x}{x^2 - 4}$$. Denominator $$0^+$$. Limit: $$+\infty$$. 16. Problem 20: $$\lim_{x \to -2^+} \frac{x}{x^2 - 4}$$. Denominator $$x^2 - 4 = (x-2)(x+2)$$. At $$x=-2$$ denominator 0. For $$x \to -2^+$$, $$x+2 \to 0^+$$, $$x-2 \to -4$$. Denominator $$\to 0^-$$. Numerator $$\to -2$$. Limit: $$\frac{-2}{0^-} = +\infty$$. 17. Problem 21: $$\lim_{y \to 6^-} \frac{y+6}{y^2 - 36}$$. Denominator $$y^2 - 36 = (y-6)(y+6)$$. At $$y=6$$ denominator 0. For $$y \to 6^-$$, $$y-6 \to 0^-$$, $$y+6 \to 12$$. Denominator $$\to 0^-$$. Numerator $$\to 12$$. Limit: $$\frac{12}{0^-} = -\infty$$. 18. Problem 22: $$\lim_{y \to 6^+} \frac{y+6}{y^2 - 36}$$. Denominator $$0^+$$. Limit: $$+\infty$$. 19. Problem 23: $$\lim_{y \to 6} \frac{y+6}{y^2 - 36}$$. Left limit $$-\infty$$, right limit $$+\infty$$. Limit does not exist. 20. Problem 24: $$\lim_{x \to 4^+} \frac{3 - x}{x^2 - 2x - 8}$$. Denominator factors: $$(x-4)(x+2)$$. At 4 denominator 0. For $$x \to 4^+$$, $$x-4 \to 0^+$$. Numerator $$3 - 4 = -1$$. Denominator $$\to 0^+$$. Limit: $$\frac{-1}{0^+} = -\infty$$. 21. Problem 25: $$\lim_{x \to 4^-} \frac{3 - x}{x^2 - 2x - 8}$$. Denominator $$0^-$$. Limit: $$\frac{-1}{0^-} = +\infty$$. 22. Problem 26: Limit does not exist (left and right limits differ). 23. Problem 27: $$\lim_{x \to 2^+} \frac{1}{|2 - x|}$$. As $$x \to 2^+$$, $$2 - x \to 0^-$$, absolute value $$0^+$$. Limit: $$+\infty$$. 24. Problem 28: $$\lim_{x \to 3^-} \frac{1}{|x - 3|}$$. As $$x \to 3^-$$, $$x - 3 \to 0^-$$, absolute value $$0^+$$. Limit: $$+\infty$$. 25. Problem 29: $$\lim_{x \to 9} \frac{x - 9}{\sqrt{x} - 3}$$. Indeterminate form 0/0. Rationalize denominator: Multiply numerator and denominator by $$\sqrt{x} + 3$$: $$\frac{(x-9)(\sqrt{x} + 3)}{(\sqrt{x} - 3)(\sqrt{x} + 3)} = \frac{(x-9)(\sqrt{x} + 3)}{x - 9}$$. Cancel $$x-9$$: Limit becomes $$\lim_{x \to 9} (\sqrt{x} + 3) = 3 + 3 = 6$$. 26. Problem 30: $$\lim_{y \to 4} \frac{4 - y}{2 - \sqrt{y}}$$. Indeterminate 0/0. Multiply numerator and denominator by $$2 + \sqrt{y}$$: $$\frac{(4 - y)(2 + \sqrt{y})}{(2 - \sqrt{y})(2 + \sqrt{y})} = \frac{(4 - y)(2 + \sqrt{y})}{4 - y}$$. Cancel $$4 - y$$: Limit becomes $$\lim_{y \to 4} (2 + \sqrt{y}) = 2 + 2 = 4$$. 27. Problem 31: $$\lim_{t \to 0.5} \frac{-16t^2 + 29t - 10.5}{t - 0.5}$$. Evaluate numerator at 0.5: $$-16(0.25) + 29(0.5) - 10.5 = -4 + 14.5 - 10.5 = 0$$. Indeterminate. Use derivative or factor numerator: Derivative numerator: $$-32t + 29$$. Evaluate at 0.5: $$-16 + 29 = 13$$. Limit is 13. 28. Problem 32: $$s(t) = -16t^2 + 29t + 6$$. Find $$\lim_{t \to 1.5} \frac{s(t) - s(1.5)}{t - 1.5}$$. This is the definition of derivative at $$t=1.5$$. Derivative: $$s'(t) = -32t + 29$$. Evaluate at 1.5: $$-32(1.5) + 29 = -48 + 29 = -19$$. 29. Problem 33: $$f(x) = \begin{cases} x - 1, & x \leq 3 \\ 3x - 7, & x > 3 \end{cases}$$ (a) $$\lim_{x \to 3^-} f(x) = 3 - 1 = 2$$. (b) $$\lim_{x \to 3^+} f(x) = 3(3) - 7 = 9 - 7 = 2$$. (c) Since left and right limits equal 2, $$\lim_{x \to 3} f(x) = 2$$. 30. Problem 34: $$g(t) = \begin{cases} t^2, & t \geq 0 \\ t - 2, & t < 0 \end{cases}$$ (a) $$\lim_{t \to 0^-} g(t) = 0 - 2 = -2$$. (b) $$\lim_{t \to 0^+} g(t) = 0^2 = 0$$. (c) Limits differ, so $$\lim_{t \to 0} g(t)$$ does not exist. 31. Problem 35: $$f(x) = \frac{x^3 - 1}{x - 1}$$. (a) Factor numerator: $$(x-1)(x^2 + x + 1)$$. Simplify: $$x^2 + x + 1$$. Evaluate at 1: $$1 + 1 + 1 = 3$$. (b) Graph is a parabola shifted, continuous except at $$x=1$$ where original function is undefined but limit exists. 32. Problem 36: $$f(x) = \begin{cases} \frac{x^2 - 9}{x + 3}, & x \neq -3 \\ k, & x = -3 \end{cases}$$ (a) Factor numerator: $$(x-3)(x+3)$$. Simplify for $$x \neq -3$$: $$x - 3$$. Limit as $$x \to -3$$: $$-3 - 3 = -6$$. Set $$k = -6$$. (b) With $$k = -6$$, $$f(x)$$ is continuous and equals $$x - 3$$ for all $$x \neq -3$$, and $$f(-3) = -6$$. 33. Problem 37: (a) Incorrect because $$\lim (1/x - 1/x^2) \neq \lim 1/x - \lim 1/x^2$$ when limits are infinite. (b) $$\lim_{x \to 0^+} (1/x - 1/x^2) = +\infty - +\infty$$ is indeterminate. Rewrite: $$\frac{1}{x} - \frac{1}{x^2} = \frac{x - 1}{x^2}$$. As $$x \to 0^+$$, numerator $$\to -1$$, denominator $$\to 0^+$$. Limit: $$-\infty$$. 34. Problem 38: $$\lim_{x \to 0^+} (1/x + 1/x^2)$$. Both terms $$\to +\infty$$. Sum $$\to +\infty$$. Total problems solved: 34 (from 5 to 38).