LIMIT

- Kelvin Asclepius Minor –

1 1−√𝑥
1. Find lim ! 21. Find lim !
𝑥→0 𝑥 𝑥→1 1−𝑥

1
2. Find lim ! 𝑥 2 −25
𝑥→0 𝑥 2 22. Find lim !
𝑥→5 𝑥−5

3. Find lim √𝑥 ! 𝑥 2 +𝑥−12

𝑥→0 23. Find lim !
𝑥→3 𝑥−3
1
4. Find lim ! 1 1
+
𝑥→0 √𝑥 𝑥 2
24. Find lim !
𝑥→−2 𝑥 3 +8
1
5. Find lim !
𝑥→∞ 𝑥 𝑥 4 −1
25. Find lim !
𝑥→1 𝑥 5 −1
𝑥
6. Find lim 𝑒 !
𝑥→∞
𝑥−4
26. Find lim !
𝑥 𝑥→4 √𝑥−2
7. Find lim 𝑒 !
𝑥→−∞
𝑥−3
27. Find lim !
2 𝑥 𝑥→3 2𝑥−6
8. Find lim ( ) !
𝑥→∞ 5
(𝑥−2)2 −1
28. Find lim !
𝑥→3 𝑥−3
−𝑥 + 3 , 𝑤ℎ𝑒𝑛 𝑥 < 2
9. 𝑓(𝑥) = {
√𝑥 − 2 + 1 , 𝑤ℎ𝑒𝑛 𝑥 ≥ 2 𝑥
29. Find lim !
a. Find lim 𝑓(𝑥) ! 𝑥→0 √𝑥
𝑥→2
b. Find lim 𝑓(𝑥) ! 𝑥 2 −𝑥−2
𝑥→6 , 𝑤ℎ𝑒𝑛 𝑥 ≠ 2
3−𝑥 , 𝑤ℎ𝑒𝑛 𝑥 < 1 30. 𝑓(𝑥) = { 𝑥−2

1 , 𝑤ℎ𝑒𝑛 𝑥 = 1 1 , 𝑤ℎ𝑒𝑛 𝑥 = 2
10. 𝑓(𝑥) = {
Find lim 𝑓(𝑥) !
2 + √𝑥 − 1 , 𝑤ℎ𝑒𝑛 𝑥 > 1 𝑥→2
a. Find lim 𝑓(𝑥) ! 31. Find lim (3𝑥 2 + 2𝑥 + 1) !
𝑥→0 𝑥→∞
b. Find lim 𝑓(𝑥) !
𝑥→1 4𝑥 2 +3𝑥+5
2 32. Find lim !
11. Find lim(8 − 3𝑥 + 12𝑥 ) ! 𝑥→∞ 7𝑥 2 +2𝑥+1
𝑥→2

6+4𝑥 𝑥 4 +3𝑥 2 +1
12. Find lim ! 33. Find lim !
𝑥→∞ 𝑥 2 +5
𝑥→−3 𝑥 2 +1

𝑥−1 𝑥 3 +𝑥 2 +1
13. Find lim ! 34. Find lim
𝑥→0 𝑥+2 𝑥→∞ 2𝑥+𝑥 4

𝜋 3+2𝑥
14. Find lim 2sin( ) ! 35. Find lim !
𝑥→4 𝑥 𝑥→∞ 2+3𝑥

𝜋𝑥 𝑥 2 −9
15. Find lim cos( ) ! 36. Find lim !
𝑥→1 3
𝑥→3 √𝑥 2 +7−4

16. Find lim sec(2𝑥) ! √2𝑥−1−√3𝑥−2

𝑥→0 37. Find lim !
𝑥→1 2𝑥−2
17. Find lim ln(𝑥) !
𝑥→0 𝑥 2 −𝑥−6
38. Find lim !
𝑥→3 4−√5𝑥+1
18. Find lim arctan(𝑥) !
𝑥→∞
(𝑥 3 +𝑥−10)(𝑥−√𝑥+2)
39. Find lim 2 !
2𝑥 2 𝑥→2 (√3𝑥+10−4)
19. Find lim !
𝑥→1 𝑥 2 −1
40. Find lim (√𝑥 2 + 1 − 𝑥) !
2𝑥 2 𝑥→∞
20. Find lim !
𝑥→−1 𝑥 2 −1

- Kelvin Asclepius Minor -

 LIMIT
- Kelvin Asclepius Minor –

1 1 2 1 2 3
1 𝑥 1 (1− )
𝑥
(1− )(1− )
𝑥 𝑥
(1− )(1− )(1− )
𝑥 𝑥 𝑥
41. If (1 + ) = 1 + + + + +⋯
𝑥 1! 2! 3! 4!
𝑥2 𝑥3 𝑥4
and 𝑒 𝑥 = 1 + 𝑥 + + + +⋯
2! 3! 4!
1 𝑥 1 𝑥
Find lim (1 + ) and lim (1 + ) !
𝑥→∞ 𝑥 𝑥→−∞ 𝑥

1
𝑥−5 3−2𝑥
42. Find lim(1 + 𝑥)𝑥 ! 47. Find lim ( ) !
𝑥→0 𝑥→∞ 𝑥−4

𝑥
1 4𝑥−20 3−2𝑥
43. Find lim (1 − ) ! 48. Find lim ( ) !
𝑥→∞ 4𝑥 𝑥→∞ 𝑥−4

1
1+𝑥 4𝑥 𝑒 𝑥 −1
44. Find lim ( ) ! 49. Find lim !
𝑥→0 𝑥
𝑥→0 1−𝑥

2𝑥
2𝑥+3 3𝑥 50. Find lim !
45. Find lim ( ) ! 𝑥→0 4 𝑥 −1
𝑥→∞ 2𝑥−1

2𝑥+3 3𝑥+4
46. Find lim ( ) !
𝑥→∞ 2𝑥−1

sin (4𝑥)
51. If √5 − 2𝑥 2 ≤ 𝑓(𝑥) ≤ √5 − 𝑥 2 56. Find lim !
𝑥→0 4x

Find lim 𝑓(𝑥) ! 𝑥

𝑥→0 57. Find lim !
𝑥→0 sin (𝑥)
2−cos (𝑥)
52. Find lim ! sin (4𝑥)
𝑥→∞ 𝑥+3 58. Find lim !
𝑥→0 2𝑥
5𝑥 2 −sin (3𝑥)
53. Find lim ! sin (5𝑥)
𝑥→∞ 𝑥 2 +10 59. Find lim !
𝑥→0 tan (2𝑥)
1
54. Find lim 𝑥 2 cos ( ) ! 4𝑥+3 sin(𝑥)+4tan (2𝑥)
𝑥→0 𝑥 60. Find lim 𝑥 !
𝑥→0 𝑥+2sin ( )
2
sin(𝑥) 𝑥3 𝑥5 𝑥7 𝑥9
55. Find lim ! (sin(𝑥) = 𝑥 − + − + − ⋯)
𝑥→0 𝑥 3! 5! 7! 9!

cos(𝑥)−1 1 1
61. Find lim ! 66. Find lim( − )!
𝑥→0 𝑥 𝑥→0 𝑥 𝑒 𝑥 −1

62. Find lim

cos(𝑥)−1
! 67. Find lim 𝑥𝑒 𝑥 !
𝑥→−∞
𝑥→0 sin(𝑥)
1
tan(𝑥−2) 68. Find lim 𝑥 𝑥 !
63. Find lim ! 𝑥→∞
𝑥→2 𝑥 2 −4

sin(𝑥)−cos(𝑥) 69. Find lim 𝑥 ln(𝑥) !

64. Find lim𝜋 2 2 ! 𝑥→0
𝑥→ cos (𝑥)−sin (𝑥)
4
70. Find lim 𝑥 sin (𝑥) !
(𝑥+sin(𝑥))4 𝑥→0
65. Find lim !
𝑥→0 𝑥4

- Kelvin Asclepius Minor -

 LIMIT
- Kelvin Asclepius Minor –

71. Find an equation of the tangent line to the parabola 𝑦 = 𝑥 2 at the point P (1, 1) !
1
72. Find an equation of the tangent line to the parabola 𝑦 = at the point P (2, -1) !
1−𝑥

73. A ball is thrown into the air, and its height t seconds later is given by 𝑠 = 40𝑡 − 16𝑡 2 𝑓𝑒𝑒𝑡. What is the (instantaneous)

velocity of the ball when it is thrown into the air ?

74. A ball is thrown into the air, and its height t seconds later is given by 𝑠 = 40𝑡 − 16𝑡 2 𝑓𝑒𝑒𝑡. What is the (instantaneous)

velocity of the ball after 2 seconds ?

𝑛2 +𝑛
75. If ∑𝑛𝑥=1 1 = 𝑛 and ∑𝑛𝑥=1 𝑥 = , find the area under 𝑓(𝑥) = 4𝑥 − 1 and above 𝑥 − 𝑎𝑥𝑖𝑠 from 𝑥 = 1 to 𝑥 = 4 using infinite
2

rectangles!

- Kelvin Asclepius Minor -