Limits · CAPE Pure Mathematics Unit 1

85 past-paper questions on Limits, part of Calculus I, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

1. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1In the diagram above showing the graph of y^2 = x, y is NOT defined for
2. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The function f(x) = \frac{x^2 - 2}{x + 2} is discontinuous for the domain value of
3. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1\lim_{x \to 3} \frac{x^2 - 9}{x - 3} is
4. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Given that \lim_{x \to 0} \frac{\sin x}{x} = 1, where x is measured in radians, then the value of \lim_{x \to 0} \frac{\sin 4x}{x} is
5. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that f(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ ax + 2 & \text{for } x \ge 3. \end{cases} For the function to be continuous at x = 3, the value of 'a' should be
6. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The value of \lim_{x \to 0} \frac{e^{2x} - 1}{e^x - 1} is
7. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1A dial on a plane preparing for landing registers the number 200 + 5\left(\frac{\sin h}{h}\right), where h is the height above the ground. Just as the plane lands the dial reads
8. 5(a)5 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine \lim_{x \to 2} \frac{x^2 + 3x - 10}{x^2 + x - 6}.
9. 11(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Use the result that (sqrt(x + h) + sqrt(x))(sqrt(x + h) - sqrt(x)) = h to show that lim_{h -> 0} (sqrt(x + h) - sqrt(x)) / h = 1 / (2 sqrt(x)).
10. 12(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the real values of x for which the function f(x) = x / (x^2 - 2x - 8) is discontinuous.
11. 11(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find lim_(x -> 1) (x^2 + x - 2)/(x^2 - 3x + 2).
12. 11(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the values of x in R such that the function f(x) = (9 - x^2)/((x^2 - 3)(|x| - 3)) is discontinuous.
13. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Based on the diagram above, which of the following statements is NOT correct?
14. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Given that \lim_{x \to 0} \frac{\sin x}{x} = 1, where x is measured in radians, then \lim_{x \to 0} \frac{\sin 3x}{2x} is
15. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The function g is defined as g(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ px + 2 & \text{for } x \ge 3. \end{cases} For the function to be continuous at x = 3, the value of 'p' should be
16. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Item 31 refers to the graph below of a function y = f(x). Based on the graph above, \lim_{x \to 1^+} f(x) =
17. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Which of the following real functions is discontinuous?
18. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The value of \lim_{x \to 0} \frac{\sin 3x}{x} is
19. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\lim_{x \to 2} \frac{x^3 - 2^3}{x - 2} =
20. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Given that \lim_{x \to 0} \frac{\sin x}{x} = 1, where x is measured in radians, then \lim_{x \to 0} \frac{\sin 3x}{2x} is
21. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The function g is defined as g(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ px + 2 & \text{for } x \ge 3 \end{cases} For the function to be continuous at x = 3, the value of 'p' should be
22. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Based on the graph above, \lim_{x \to 1^+} f(x) =
23. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The function g is defined as g(x) = \begin{cases} 3x + 5, & x < 3 \\ px + 2, & x \ge 3 \end{cases}. For the function to be continuous at x = 3 the value of p should be
24. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The value of \lim_{x \to 0} \frac{\sin 3x}{x} is
25. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Given that \lim_{x \to 0} \frac{\sin x}{x} = 1, where x is measured in radians, then \lim_{x \to 0} \frac{\sin 3x}{2x} is
26. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1In the diagram above showing y^2 = x, y is NOT defined for
27. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1\lim_{x \to 3} \frac{x^2 - 9}{x - 3} is
28. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Given that \lim_{x \to 0}\frac{\sin x}{x} = 1, where x is measured in radians, then \lim_{x \to 0}\frac{\sin 3x}{2x} is
29. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The function g is defined as g(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ px + 2 & \text{for } x \ge 3 \end{cases} For the function to be continuous at x = 3, the value of 'p' should be
30. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1\lim_{x \to -5} \frac{x + 5}{x^2 - 25} =
31. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The value of \lim_{x \to 0} \frac{\sin 3x}{x} is
32. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Given that f(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ ax + 2 & \text{for } x \ge 3 \end{cases} For the function to be continuous at x = 3, the value of a should be
33. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1\lim_{x \to 2} \frac{x^3 - 2^3}{x - 2} =
34. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1\lim_{x \to -1} \frac{x^2 - x - 2}{x - 1} =
35. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1A dial on a plane preparing for landing registers the number 200 + 5\left(\frac{\sin h}{h}\right), where h is the height above the ground. Just as the plane lands the dial reads
36. 5(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Determine \lim_{x \to \infty} \frac{2x^3 - 4x + 1}{3x^4 + x^2 - 2}.
37. 5(a)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Determine lim_{x -> -2} f(x).
38. 5(a)(iii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Hence, or otherwise, determine whether f is continuous at x = -2.
39. 5(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Evaluate lim_{theta -> 0} (sin theta / sin 4 theta).
40. 5(a)(i)a)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the value of lim_{x → -5⁺} f(x).
41. 5(a)(i)b)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the value of lim_{x → 5} f(x).
42. 5(a)(ii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2State the value of k such that lim_{x → k} f(x) = -2.
43. 5(b)(i)b)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine lim_{x → 1} g(x).
44. 5(b)(ii)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Hence, state whether g is continuous at x = 1. Give a reason for your answer.
45. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)If f is continuous at x = 0, determine the value of a.
46. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)Find lim (as x→3) (x² - 27) / (x² + x - 12).
47. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)Determine the value of k for which f(x) = (x²-1)/(x-1) for x ≠ 1 and f(x) = k for x = 1 is continuous for all values of x.
48. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)Evaluate lim (x²-2x-3)/(x²-4x+3) as x approaches 3.
49. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)Find lim (as x→2) (x² + 5x + 6) / (x² - x - 6).
50. 5(a)5 marks· Pure Mathematics · Unit 1 Q5 5(a)By expressing x - 4 as (√x + 2)(√x - 2), find lim (x→4) (√x - 2) / (x - 4). Hence, find lim (x→4) (√x - 2) / (x² - 5x + 4).
51. 5(a)3 marks· Pure Mathematics · Unit 1 Q5 5(a)Use L'Hopital's rule to obtain lim (x→0) (sin 4x / sin 5x).
52. 5(a)5 marks· Pure Mathematics · Unit 1 Q5 5(a)Find lim (x³ - 8) / (x² - 6x + 8) as x → 2.
53. 5(a)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(i)Find lim (x→2) f(x).
54. 5(a)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(i)Find lim x→3 of (x²-9) / (x³-27).
55. 5(a)(i)1 mark· Pure Mathematics · Unit 1 Q5 5(a)(i)State the value of lim (as δx → 0) (sin 8x / δx).
56. 5(a)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(i)Find the value of a if f(x) is continuous at x = 3.
57. 5(a)(i)1 mark· Pure Mathematics · Unit 1 Q5 5(a)(i)State the value of lim(u→0) (sin u)/u.
58. 5(a)(i)2 marks· Pure Mathematics · Unit 1 Q5 5(a)(i)Find the values of x for which (x³+8)/(x²-4) is discontinuous.
59. 5(a)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)Determine whether f(x) is continuous at x = 2. Give a reason for your answer.
60. 5(a)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)By means of the substitution u = 3x, show that lim(x→0) (sin 3x)/x = 3.
61. 5(a)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)Find lim x→0 of (tan x - 5x) / (sin 2x - 4x).
62. 5(a)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)Hence, or otherwise, find lim_{x→-2} (x³+8)/(x²-4).
63. 5(a)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)find the value of b.
64. 5(a)(iii)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(iii)Hence, evaluate lim(x→0) (sin 3x)/(sin 5x).
65. 5(a)(iii)5 marks· Pure Mathematics · Unit 1 Q5 5(a)(iii)By using the fact that lim_{x→0} (sin x)/x = 1, or otherwise, find, lim_{x→0} (2x²+4x)/(sin 2x).
66. 5(b)3 marks· Pure Mathematics · Unit 1 Q5 5(b)Determine the values of x ∈ R for which the function (x+2)/(x(x+1)) is NOT continuous.
67. 5(b)1 mark· Pure Mathematics · Unit 1 Q5 5(b)Find
68. 5(b)(i)a)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)a)Find lim x→4⁺ of f(x).
69. 5(b)(i)a)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)a)lim_{x→1+} f(x)
70. 5(b)(i)b)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)b)Find lim x→4⁻ of f(x).
71. 5(b)(i)b)4 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)b)the value of the constant p such that lim_{x→1} f(x) exists.
72. 5(b)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Determine lim (as x→2⁺) f(x)
73. 5(b)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Deduce that f(x) is discontinuous at x = 4.
74. 5(b)(ii)1 mark· Pure Mathematics · Unit 1 Q5 5(b)(ii)Hence, determine the value of f(1) for f to be continuous at the point x = 1.
75. 5(b)(ii)a)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)a)Find lim f(x) as x → 1+.
76. 5(b)(ii)b)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)b)Find lim f(x) as x → 1-.
77. 5(b)(iii)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(iii)Determine lim (as x→2⁻) f(x) in terms of the constant b
78. 5(b)(iii)3 marks· Pure Mathematics · Unit 1 Q5 5(b)(iii)Deduce that f(x) is continuous at x = 1.
79. 5(b)(iv)4 marks· Pure Mathematics · Unit 1 Q5 5(b)(iv)Determine the value of b such that f is continuous at x = 2.
80. 6(a)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Determine whether or not the lim (x→1) f(x) exists.
81. 6(a)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Determine whether or not the limit of f at x = 1 exists.
82. 6(a)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Determine whether f is continuous at x = 1.
83. 6(a)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Determine whether f is continuous at x = 1.
84. 6(b)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Calculate the lim (x→0⁻) f(x) and lim (x→0⁺) f(x).
85. 6(b)(ii)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Hence, determine the values of a and b such that f(x) is continuous at x = 0.