Catalog of limits

Question

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For more comprehensive catalog see https://mathindex.wordpress.com (it includes more limits, series, finite sums, and recurrences).

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This is an attempt to organize questions about evaluation of common limits. Discussed at Catalog of standard exercises

Trigonometric limits

• with sine, at $0$
• with sine, at $\infty$
• with sine, at points $\ne 0,\infty$
• with cosine (no sine), at $0$
• with cosine (no sine), at $\ne 0$

Answer 1

Trigonometric limits: with sine, at $0$

1. $\displaystyle \lim\limits_{x\to 0}\frac{\sin x}x=1$ 1, 2, 3, 4, 5, 6, 7, 8.
2. $\displaystyle \lim_{x \to 0} (1+ \sin 2x)^{\frac{1}{x}}$
3. $\displaystyle \lim_{(x,y)\to(0,0)}\frac{(\sin^2x)(e^y-1)}{x^2+3y^2}$
4. $\displaystyle \lim_{r \to 0} \frac{r^3 \cos^2\theta\sin\theta}{r^2(r^2\cos^4\theta + \sin^2\theta)}$
5. $\displaystyle \lim_{x\to 0^+} \frac{\ln(x)}{1/\sin(x)}$
6. $\displaystyle \lim_{x \to 0} \frac{x^2}{x+\sin (\frac 1 x)} $
7. $\displaystyle \lim\limits_{x \to 0^+} \ln[(1-\sin^2(x))^\frac{1}{2x}]$
8. $\displaystyle \lim_{x\to0}\frac{\sin(x)\cos(4x)}{x+x\cos(5x)} $
9. $\displaystyle \lim\limits_{x\to 0} \frac{x - \sin(x)}{x^2}$
10. $\displaystyle \lim_{x\to 0}{\frac{\frac{\sin^2{x}}{x^2}\cdot\frac{1}{\cos^2{x}} - 1}{2x}}$
11. $\displaystyle \lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$
12. $\displaystyle \displaystyle\lim_{x\to 0} \frac{\ln(1-x)-\sin x}{1-\cos^2 x}$
13. $\displaystyle \lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$
14. $\displaystyle \displaystyle\lim_{x\to 0}\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$
15. $\displaystyle \lim_{x \to 0}{\frac{x-\sin{x}}{x^3}}=\frac{1}{6}$ 1, 2
16. $\displaystyle \lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$
17. $\displaystyle \lim_{x \to 0} \frac{\sin(1/x)}{\sin(1/x)}$
18. $\displaystyle \mathop {\lim }\limits_{x \to 0} \frac{{\sin (\sin x)}}{x}?$
19. $\displaystyle \lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$
20. $\displaystyle \lim _{x\rightarrow 0}{\frac {\cos \left( x \right) \sin \left( x \right) -x}{ \left( \sin \left( x \right) \right) ^{3}}}$
21. $\displaystyle \lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$
22. $\displaystyle \displaystyle \lim_{x \to 0} \frac{\sin(\tan x) - \tan(\sin x)}{\arcsin(\arctan x) - \arctan(\arcsin x)}$
23. $\displaystyle \lim_{x\to 0}\frac{\sin^2{x^{2}}}{x^{2}}$
24. $\displaystyle \displaystyle\lim_{x\to 0+} \left(\frac{\sin x}x\right)^{1/{x^2}}$
25. $\displaystyle \lim_{x\to 0}\frac{\sin(2x)\sin(4x)}{x\sin(3x)}=\frac83$
26. $\displaystyle \lim_{x\rightarrow 0} \sin(\frac{1}{x})$
27. $\displaystyle \lim_{x \to 0} \left( \frac{\tan (\sin (x))-\sin (\tan (x))}{x^7} \right).$
28. $\displaystyle \lim_{x\to 0}\frac{\sin{6x}}{\sin{2x}}$
29. $\displaystyle \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\bigg) $
30. $\displaystyle \displaystyle\lim_{x \to 0} \dfrac{e^{\sin2x}-e^{\sin x}}{x}$
31. $\displaystyle \lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$
32. $\displaystyle \lim_{x\to0} \left\lfloor\frac{x^2}{\sin x \tan x}\right\rfloor$
33. $\displaystyle \mathop {\lim }\limits_{x \to {0^ + }} {\left( {\frac{{\sin x}}{x}} \right)^{\frac{1}{x}}}$
34. $\displaystyle \lim_{x \to 0} x\cdot \sin\left(\dfrac{1}{x}\right)=0$
35. $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{5x} = \frac{\sin(0)}{0} = \frac{0}{0} $
36. $\displaystyle \lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$
37. $\displaystyle \lim_{x\to 0} \dfrac{1-\cos^3 x}{x\sin2x}.$
38. $\displaystyle \lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$
39. $\displaystyle \lim_{x\to 0}\frac{\sin{3x}}{x}$
40. $\displaystyle \lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3x}$
41. $\displaystyle \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} = -2$
42. $\displaystyle \lim_{\theta \rightarrow 0}\frac {\sin^2\theta}{\theta}$
43. $\displaystyle \lim_{x\to 0}\frac{\cos 3x-\cos x}{x^2} = \lim_{x\to 0}\frac{-2\sin\frac{1}{2}(3x+x)\sin\frac{1}{2}(3x-x)}{x^2}=\lim_{x\to 0}\frac{-2\sin2x\sin x}{x^2}=\lim_{x\to 0}\frac{-2(2\sin x\cos x)\sin x}{x^2}=\lim_{x\to 0}\frac{-4\sin^2 x\cos x}{x^2}$
44. $\displaystyle \lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$
45. $\displaystyle \displaystyle\lim_{x\to 0}\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right) $
46. $\displaystyle \lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}}}{r^4\cos^4 \theta+r^4 \sin^4 \theta } $
47. $\displaystyle \lim \limits_{x\to 0} \frac {\tan(2x)}{\sin(x)}$
48. $\displaystyle \lim_{x \to 0}\frac{2x^2}{\sin^2 x}=2$
49. $\displaystyle \lim_{x\to 0}\frac{\sin 2x}{ x}$
50. $\displaystyle \lim_{(x,y)\rightarrow(0,0)}\frac{\sin(x)\sin(y)}{x^2+y^2}$
51. $\displaystyle \lim_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}? $
52. $\displaystyle \lim_{x \to 0} \frac{\tan(3x^2) + \sin^2(5x)}{x^2}$
53. $\displaystyle \lim_{x \to 0} \sin(2x) = 0$
54. $\displaystyle \lim_{x \to 0} \Bigl(\frac{\sin{x}}{x}\Bigr)^{1/x^{3}}$
55. $\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x} $
56. $\displaystyle \lim_{x\to0}\sin\left(\frac1x\right)$
57. $\displaystyle \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$
58. $\displaystyle \lim_{x\to0}\frac{\sin5x}{\sin4x}$
59. $\displaystyle \lim_{x\to 0} \left(\frac{x \tan x^2}{\cos 5x \sin^3 3x}\right) $
60. $\displaystyle \lim\limits_{x\to 0} \frac{\sqrt{x}}{\sin x}$
61. $\displaystyle \lim_{x\to0^{+}}\dfrac{\tan{(\tan{x})}-\tan{(\sin{x})}}{\tan{x}-\sin{x}}$
62. $\displaystyle \lim_{x\to0} \frac{\cos x + \cos 2x + \dots+ \cos nx - n}{\sin x^2}$
63. $\displaystyle \displaystyle\lim_{x\to 0}\frac{\tan x - \sin x}{x^n}$
64. $\displaystyle \lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$
65. $\displaystyle \lim_{x \to 0} \sin(x) = 0$
66. $\displaystyle \lim\limits_{x\to 0} \frac{\sin x}{x}^{\frac{1}{1-\cos x}} = \lim\limits_{x\to 0}\exp\left( \frac{\ln(\frac{\sin x}{x})}{1-\cos x} \right)$
67. $\displaystyle \lim_{x \to 0} \left(\dfrac{\sin x}{x}\right)^{\dfrac{1}{1 - \cos x}}$
68. $\displaystyle \lim_{x \rightarrow 0}\left (\frac 1x- \frac 1{\sin x} \right ) $
69. $\displaystyle \lim_{x \to 0} \frac{1}{\sqrt{x^3}} - \frac1{\sin x}$
70. $\displaystyle \lim_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$
71. $\displaystyle \lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-\cos(x)}{\sin^2(x)}\right)$
72. $\displaystyle \lim _{x \rightarrow 0} \left(\frac{ \sin x}{x}\right)^{1/x}$
73. $\displaystyle \lim\limits_{x\rightarrow 0^+}{\dfrac{e^x-\sin x-1}{x^2}}$
74. $\displaystyle \lim \limits_{x\to0} (\sin x)^x $
75. $\displaystyle \lim_{x\to 0}\frac{x^2-x}{\sin3x} $
76. $\displaystyle \lim_{\theta \to 0^+}\frac{\sin\theta}{\theta^2}?$
77. $\displaystyle \lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$
78. $\displaystyle \lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$
79. $\displaystyle \lim_{x\to0}\frac{\sin(1-\cos(x))}{x^2e^x}$
80. $\displaystyle \lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)=\lim_{x\to 0} \cos \bigg(\pi x^2 (\frac {\cos (6x)} {\sin (\frac {x} {2}) \sin 6x} \bigg)$
81. $\displaystyle \lim_{x\to0}\frac{\sqrt{5x+3}-\sqrt 3}{5^{\sin(7x)}-1}$
82. $\displaystyle \lim_{x\to0} \frac{\tan(\tan x) - \sin(\sin x)}{\tan x -\sin x}$
83. $\displaystyle \lim_{x \rightarrow 0}\sin(x)\ln{\sin{x}}$
84. $\displaystyle \lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$
85. $\displaystyle \lim_{r\rightarrow0}\frac{\cos\theta·\sin(r^2)}{r}=\lim_{r\rightarrow0}\frac{\cos\theta·r^2}{r}=\lim_{r\rightarrow0}{\cos\theta·r}=0 \text{ for all }\theta \in [0,2\pi)$

Answer 2

Trigonometric limits: with sine, at points $\ne 0,\infty$

1. $\displaystyle \lim\limits_{x\to a}\sin x=\sin a$
2. $\displaystyle \lim x\sin (1/x) = \lim\, x \dfrac{\sin (1/x)}{x(1/x)} = \lim\, x/x = 1$
3. $\displaystyle \lim_{x\to\pi} \frac{\sin5x}{\sin3x}$
4. $\displaystyle \lim_{x\to 1}\frac{(x^2-1)\sin(3x-3)}{\cos(x^3-1)\tan^2(x^2-x)}.$
5. $\displaystyle \lim_{x\to\pi/2} \frac{\sin(x)}{x} = \lim_{x\to\pi/2} \frac{\cos(x)}{1}=0$
6. $\displaystyle \lim_{x\to\pi/2}(\tan x)(\ln \sin x)$
7. $\displaystyle \lim_{x\rightarrow\frac{\pi}{4}}\frac{\cos(2x)}{\cos(x)-\sin(x)}=\lim_{x\rightarrow\frac{\pi}{4}}\frac{2\cos^{2}(x)-1}{\cos(x)-\sqrt{1-\cos^{2}(x)}}$
8. $\displaystyle \lim \limits _{x \rightarrow 1} \dfrac{\sin (\pi x^{\alpha})}{\sin (\pi x^{\beta})}$
9. $\displaystyle \lim_{x\to2^-}h(x)=\lim_{x\to2^-}\sin k(x)=\sin(2k)$
10. $\displaystyle \lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$
11. $\displaystyle \lim_ {x \to 1} \frac{\sin{\pi x}}{1 - x^2} $
12. $\displaystyle \lim_{x \to \frac{\pi}{2}} \frac{\sin^2x-1}{\sin x-1}$
13. $\displaystyle \lim_{x \to \frac{\pi}{6}}\frac{2\sin{(x)}-1}{\sqrt{3}\tan{(x)}-1}$
14. $\displaystyle \lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{1-\sqrt{2}\sin x}$
15. $\displaystyle \displaystyle \lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4 $
16. $\displaystyle \lim_{x \to \pi/2} \frac{1 - \sin{x}}{(2x - \pi)^2}$
17. $\displaystyle \lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$
18. $\displaystyle \lim_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$
19. $\displaystyle \lim_{x\to 1} \frac{\sin(x^2-1)}{x-1}$
20. $\displaystyle \lim_{x \to \frac{\pi}{4}}\frac{\sin x-\cos x}{\ln(\tan x)}=\left(\lim_{x \to \frac{\pi}{4}}\frac{\sin x-\cos x}{\ln(\tan x)}\right):\cos x = \lim_{x \to \frac{\pi}{4}}\frac{\tan x-1}{\frac{\ln(\tan x)}{\cos x}} = \frac{0}{\frac{0\cdot{2}}{0\cdot\sqrt{2}}}$
21. $\displaystyle \lim\limits_{x\to\pi}\left(\dfrac1{\sin(x)}\right)^{x-\pi}$
22. $\displaystyle \lim_{x\to\pi/2}\frac{\sin x - (\sin x)^{\sin x}}{1 - \sin x + ln(\sin x)}$
23. $\displaystyle \lim_{x\to\pi/6}\frac{1 - 2\sin{x}}{2\sqrt{3}\cos{x} - 3} $
24. $\displaystyle \lim_{x \to \pi}\frac{e^{\sin x} - 1}{x - \pi}$
25. $\displaystyle \lim \frac{\sin(h)}{h}=1,\ \mbox{as}\ h\to0$
26. $\displaystyle \lim_{x\to \pi}\sin(x + \sin x)$
27. $\displaystyle \lim_{x\to\pi}\frac{\sin(3x)}{\sin(2x)}$
28. $\displaystyle \lim_{x\rightarrow 1}\frac{\sin{(x^2-1)}}{x-1}$
29. $\displaystyle \lim_{x\to \sqrt{n}^+} \frac{n\sin^2(x\pi)-n\sin^2(\sqrt{n}\pi)}{x-\sqrt{n}} = n\pi\sin(2\pi\sqrt{n})$
30. $\displaystyle \lim\limits_{x \to \pi/2} \tan^2(x) = \lim\limits_{x \to \pi/2} \frac{\sin^2(x)}{\cos^2(x)} = \lim\limits_{x \to \pi/2} \frac{1 - \cos^2(x)}{1-\sin^2(x)} = \infty$
31. $\displaystyle \lim_{x \to \pi} \frac{\cos2(x-\pi)}{\sin2(x-\pi)}\frac{\cos(x-\frac\pi2)}{\sin(x-\frac\pi2)} = \lim_{x \to \pi} \frac{-\cos2x}{-\sin2x}\frac{-\sin x}{-\cos x} = \lim_{x \to \pi} \frac{\cos2x}{2\sin x\cos x}\frac{\sin x}{\cos x} = \frac12\lim_{x \to \pi} \frac{\cos2x}{\cos x}\frac{1}{\cos x}=\frac12 $
32. $\displaystyle \displaystyle \lim\limits_{x \to {\pi/2}} \frac {\sin x -(\sin x)^{\sin x}} {1-\sin x+\log (\sin x)}$
33. $\displaystyle \lim_{x\rightarrow -\pi}\large{\frac {\sin(4x)}{x^2+\pi x}}$
34. $\displaystyle \lim_{x \rightarrow \pi/4} \frac{1-\tan x }{1-\sqrt2 \, \sin x}$
35. $\displaystyle \lim_{x \to \pi/ 6} \frac{(2\sin x + \cos(6x))^2}{(6x - \pi)\sin(6x)}$
36. $\displaystyle \displaystyle\lim_{x \rightarrow \frac{\pi}{6}} (2+\cos {6x})^{\ln |\sin {6x}|}$
37. $\displaystyle \lim_{x \rightarrow 1} \left((x - 1)\sin\left(\frac{x}{x-1}\right)\right)$
38. $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{2{{x}^{4}}+{{x}^{3}}+{{x}^{2}}\sin x}{{{x}^{2}}-5{{x}^{4}}+{{x}^{3}}\sqrt{x}}$
39. $\displaystyle \lim_{x \rightarrow a} \left( \frac{\sin x}{\sin a} \right)^{\frac{1}{x-a}}$
40. $\displaystyle \lim_{x\to a}[\cos(2 \pi x)-\sin(2 \pi x) \cot(\frac{\pi x}{a})]$
41. $\displaystyle \lim_{x\to π/2} \frac{1-\sin x+\cos x}{\sin 2x -\cos x}$
42. $\displaystyle \lim_{x\to \pi} \dfrac{\sin(3x)}{\sin(5x)}$
43. $\displaystyle \lim_{x\to 1} \frac{\sin (x-1)}{x-1}$
44. $\displaystyle \lim_{x \to \frac{\pi}{2}}\log y=\lim_{x \to \frac{\pi}{2}}[\tan x\log(1+\cos x)]=\lim_{x \to \frac{\pi}{2}}\frac{\log(1+\cos x)}{\cos x} \times \lim_{x \to \frac{\pi}{2}} \sin x$
45. $\displaystyle \lim_{x\to ∞}e ^ {-\sin{x}}$
46. $\displaystyle \lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$
47. $\displaystyle \lim _{ { x }\to { 0 } }{ \frac { \sin x-\arctan x }{ {x }^{ 2 }\log(1+x) } }$
48. $\displaystyle \lim_{x\to\pi/2}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\log_e \sin x}=2$
49. $\displaystyle \displaystyle\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$
50. $\displaystyle \lim_{x\rightarrow 1} \dfrac{x\sin\lbrace x\rbrace}{x-1} $
51. $\displaystyle \lim_{x \to \dfrac{\pi}{4}} \dfrac{\sin x-\dfrac{\sqrt{2}}{2}}{x-\dfrac{\pi}{4}}$
52. $\displaystyle \lim_{E \to U} \frac{1}{4}\left[\frac {U^2}{E(E-U)}\right]\sin^2 k'L$
53. $\displaystyle \lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$
54. $\displaystyle \lim_{x \to a} \sin{x} = ?$
55. $\displaystyle \lim _{x\to\frac{\pi}{2}} \frac{[\frac{x}{2}]}{\log(\sin x)}$
56. $\displaystyle \lim_{x\rightarrow \frac{\pi }{4}}\frac{\sin2x-\cos^{2}2x-1}{\cos^{2}2x+2\cos^{2}x-1}$
57. $\displaystyle \lim_{x\to h^+}c_1\cos(x-a)+c_2\sin (x-a) $
58. $\displaystyle \displaystyle \lim_{x \to 1 y \to 1} (2y-x)^{\frac{1}{\sin(2y-x-1)}}$
59. $\displaystyle \lim\limits_{x\to1}{\dfrac{1-x}{1-\sin\left(\dfrac{\pi \cdot x}{2}\right)}}$

Answer 3

Trigonometric limits: with cosine (no sine), at $\ne 0$

At finite nonzero points

1. $\displaystyle \lim_{x \rightarrow a} \cos^{-1}(x)=\cos^{-1}(\lim_{x \rightarrow a} x)$
2. $\displaystyle \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} $
3. $\displaystyle \lim_{x\to \frac{\pi}{2}} \frac{\cos(x)}{\frac{\pi}{2}-x}\;$
4. $\displaystyle \lim_{x\rightarrow{\frac\pi2 }} (\sec(x) \tan(x))^{\cos(x)} $
5. $\displaystyle \lim_{x\to \pi/2} {\cos x\over x-\pi/2} $
6. $\displaystyle \lim_{x \rightarrow (-1)^{+}}\left(\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}} \right)$
7. $\displaystyle \lim_{x \rightarrow α}\frac{1 - \cos(ax^2+bx+c)}{(x-α)^2}.$
8. $\displaystyle \lim_{x\to\pi/4}\frac{1-\cos x}{x}$
9. $\displaystyle \displaystyle \lim_{x \to \pi/2} \frac{1-\cos 2x}{2\cos x}$
10. $\displaystyle \lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)$
11. $\displaystyle \lim_{x\,\to\tfrac{\pi}{6}} \frac{\cos(2x) - \frac{1}{2}}{x - \frac{\pi}{6}}$
12. $\displaystyle \displaystyle \lim_{x \rightarrow o}\frac{1-\cos x}{x^{2}}=\frac{1}{2}$
13. $\displaystyle \displaystyle \lim_{x \to \cos(x)}$
14. $\displaystyle \lim_{x \to \frac{\pi}{2}} (1+\cos x)^{\tan x}$
15. $\displaystyle \lim \sup \cos(n\pi/12)$
16. $\displaystyle \lim_{x\to1}\left(\frac{1+\cos(\pi x)}{\tan^2(\pi x)}\right)^{\!x^2}$
17. $\displaystyle \lim_{x\to 1} (1-x)^{\cos[(\Pi/2) x)]}$
18. $\displaystyle \lim_{x\to -2} \left(x-21-12\sqrt[3]{x-6}\right)^{\frac{1}{1-\cos(x+2)}}$
19. $\displaystyle \lim_{x \to a} \cos{x} = ?$
20. $\displaystyle \lim_{x\to 2} \frac{\cos(x-2)-1}{x^{2}+x-6}$
21. $\displaystyle \lim_{x\rightarrow\pi}\dfrac{1+\cos^3x}{1-\cos^2x}$
22. $\displaystyle \lim\limits_{x\to1}{\dfrac{2}{\pi \cdot \cos\left(\dfrac{\pi \cdot x}{2}\right)}} = \frac{2}{0} = \infty$

At infinity

1. $\displaystyle \lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$
2. $\displaystyle \lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$
3. $\displaystyle \lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$
4. $\displaystyle \lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots +\cos{\frac{n\pi}{n}} \right)$
5. $\displaystyle \lim \limits_{n \to \infty} \left[\cos\left(x \over 2\right)\cos\left(x \over 4\right) \cos\left(x \over 8\right)\ \cdots\ \cos\left(x \over 2^{n}\right)\right] $
6. $\displaystyle \lim_{n \rightarrow + \infty } \left(\frac{n^3}{4n-7}\right)\left(\cos\left(\frac1n\right)-1\right)$
7. $\displaystyle \lim_{x\to \infty}(e^{-x}+2\cos3x)$
8. $\displaystyle \lim_{n\to \infty } \, n \left(\frac{a_n}{a_{n+1}}-1\right)=\lim_{n\to \infty } \, n \left(\left(1-\cos \left(\frac{1}{n}\right)\right)^z \left(1-\cos \left(\frac{1}{n+1}\right)\right)^{-z}-1\right) = 2z$
9. $\displaystyle \lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac {1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot \arccos{1}}{n}$
10. $\displaystyle \lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +...+\cos (\frac{1}{n}) \right )$
11. $\displaystyle \lim_{x\to \infty} \cos^x(c)=0.7390851332$
12. $\displaystyle \lim_{n\to\infty} n\sqrt{2-2\cos\left(\frac{2\pi}{n}\right)}=2\pi $
13. $\displaystyle \lim_{x \to \infty} \, \cos \left(\dfrac{1}{x}\right)^{x} $
14. $\displaystyle \lim_{n\rightarrow \infty}(\cos{\frac{\pi}{3}n})=s$
15. $\displaystyle \lim_{n\rightarrow\infty}\sqrt{n}\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$
16. $\displaystyle \lim_{n \to \infty} n^2\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\ldots}}}$
17. $\displaystyle \lim_{n\to \infty } \,\cos (1) \cos \left(\frac{1}{2}\right) \cos \left(\frac{1}{4}\right)\cdots \cos \left(\frac{1}{2^n}\right)$
18. $\displaystyle \lim_{n \to \infty}{n \cos(\pi/2 + 1/n)}$
19. $\displaystyle \lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$
20. $\displaystyle \lim_{n\rightarrow\infty}\frac{\tan{n}}{1.5^n} = \lim_{n\rightarrow\infty}\frac{\frac{1}{\cos^2{n}}}{1.5^n\cdot\ln1.5} =\lim_{n\rightarrow\infty}\frac{1}{\ln1.5\cdot\cos^2n\cdot1.5^n} = 0$
21. $\displaystyle \lim_{n\to\infty} \cos(1)\cos(0.5)\cos(0.25)\ldots \cos(1/2^n) $
22. $\displaystyle \lim_{x\to \infty} \frac{(x+\cos x)}{x}=1$
23. $\displaystyle \lim\limits_{m\to\infty}\left(\cos\frac xm\right)^{m}$
24. $\displaystyle \lim_{n\to\infty}\frac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$
25. $\displaystyle \lim_{n \to \infty} \cos \dfrac {\pi}{2^2}\cos \dfrac {\pi}{2^3}\cos \dfrac {\pi}{2^4}......\cos \dfrac {\pi}{2^n}=\dfrac {2}{\pi}$
26. $\displaystyle \lim_{x \to \infty}x \cos (1/x) = \infty$
27. $\displaystyle \lim_{n\to\infty}\displaystyle\frac{\cos\pi n}{n^2}.$
28. $\displaystyle \lim_{x \to \infty} x\cos\frac{1}{x}=\infty$
29. $\displaystyle \lim_{n \rightarrow \infty} \left[n \; \arccos \left( \left(\frac{n^2-1}{n^2+1}\right)^{\cos \frac{1}{n}} \right)\right]$
30. $\displaystyle \lim\limits_{x \to \infty}x \cos x \neq \infty$
31. $\displaystyle \lim_{n \rightarrow \infty } \sqrt[n]{n^n+n^{n+1}+\cdots+n^{2n}} \cdot\left(1-\cos{\frac{3}{n}}\right)$
32. $\displaystyle \lim_{x \to \infty}x^{2}\ln\left(\cos\left(\pi \over x\right)\right)$
33. $\displaystyle \lim_{t\to\infty} (\cosh x)^{1/x}.$
34. $\displaystyle \lim_{x \to \infty} (\cos x)^\frac{1}{x^2} $
35. $\displaystyle \lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} $
36. $\displaystyle \lim_{n \to \infty}(n^2+1)\left(\cos\left(\dfrac{1}{n}\right)-1\right)$
37. $\displaystyle \lim _{x\to \infty \:}\left(\frac{\left(\cos ^2\left(x^3\right)+x\right)}{x!}\right)\:$
38. $\displaystyle \lim_{n\to\infty} ∫_{-1}^1f (t)\cos^2(nt)\,\mathrm dt$
39. $\displaystyle \lim\limits_{k\rightarrow \infty} \cos{kx} P_{\frac{1}{x}}$
40. $\displaystyle \lim_{n \to \infty} \left| \cos \left( \frac{\pi}{4(n-1)} \right) \right|^{2n-1} = L$
41. $\displaystyle \lim_{n \to \infty} \cos(a_n) = 1$
42. $\displaystyle \lim_{\theta\rightarrow -\infty}\dfrac{\cos\theta}{3\theta}$

Answer 4

Trigonometric limits: with sine, at $\infty$

1. $\displaystyle \lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$ 1, 2
2. $\displaystyle \displaystyle\lim_{x \to +\infty} x \sin x$
3. $\displaystyle \lim \limits_{n \to \infty} |\sin(\pi \sqrt{n^2+n+1})|$
4. $\displaystyle \lim\limits_{ x \to \infty }{ \frac { x+\sin { x } }{ x } } =1.$
5. $\displaystyle \lim_{n\to\infty}\sqrt{n}\sin_{n}(x)$
6. $\displaystyle \lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
7. $\displaystyle \lim_{n\to\infty}\sin\frac{1}{n}$
8. $\displaystyle \lim_{n \to \infty }\sqrt[n]{ \frac{\left | \sin1 \right |}{1}\cdot\frac{\left | \sin2 \right |}{2}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}} $
9. $\displaystyle \lim_{x\to\infty} \dfrac{\sin(x^2)}{x^3}$
10. $\displaystyle \lim_{n \to \infty}\big(\max \{\sin 1, \sin 2, \ldots ,\sin n\}\big) = 1?$
11. $\displaystyle \lim\limits_{x\to\infty} \frac{\ln(x^2+4)}{\sinh^{-1}x}$
12. $\displaystyle \lim_{n \to \infty} 2^n\cos\left(\frac{\pi}{2^n}\right)\sin\left(\frac{\pi}{2^n}\right)$
13. $\displaystyle \lim_{x \to \infty} 2^x \sin\left(\frac \pi {2^x}\right)=\pi.$
14. $\displaystyle \left|\left(\frac{1+\sin(k)}{2}\right)^k-\lim_{n\to\infty}\left(\frac{1+\sin(n)}{2}\right)^n\right|<\epsilon$
15. $\displaystyle \mathop{\lim}\limits_{x \to \infty}\left({x\space\sin^{2} x}\right)$
16. $\displaystyle \lim _{n\rightarrow \infty }\sin \frac {1} {3^{n}z}\rightarrow 0$
17. $\displaystyle \lim_{x \to \infty} (x/(x+1))(\sin(x^2)) = \lim_{x \to \infty} \frac{x \sin x^2 }{x+1}$
18. $\displaystyle \lim_{n\rightarrow \infty} \left(1+\sin\left(\frac {1}{n}\right)\right)^{n\cos\left(\frac {1}{n}\right)}= \lim_{n\rightarrow \infty} \left(\left(1+\sin\left(\frac {1}{n}\right)\right)^\frac {1}{\sin\left(\frac {1}{n}\right)} \right)^{n\cdot\cos\left(\frac {1}{n}\right)\sin\left(\frac{1}{n}\right)}$
19. $\displaystyle \lim\limits_{x\rightarrow\infty}{(x + 10\sin x )}$
20. $\displaystyle \lim_{n\to \infty} \frac{\tan(\pi/n)}{n\sin^2(2/n)}$
21. $\displaystyle \lim_{n\to\infty} (\sin {\frac{n}{n^2+1^2}}+\sin {\frac{n}{n^2+2^2}}+\cdots+\sin {\frac{n}{n^2+n^2}})$
22. $\displaystyle \lim\limits_{n\to\infty}{\sin(n)}$
23. $\displaystyle \lim_{n\to \infty}\sin{(n^m)}$
24. $\displaystyle \lim_{n\to\infty} \left(\frac{{\sin\frac{2}{2n}+\sin\frac{4}{2n}+\cdot \cdot \cdot+\sin\frac{2n}{2n}}}{{\sin\frac{1}{2n}+\sin\frac{3}{2n}+\cdot \cdot \cdot+\sin\frac{2n-1}{2n}}}\right)^{n}$
25. $\displaystyle \lim_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}.$
26. $\displaystyle \lim_{n \rightarrow\infty}\left(\frac{3^{-n}\sin(3^{(1-n)})}{\tan(3^{1-2n})} \right)$
27. $\displaystyle \lim_{n\rightarrow\infty}\sqrt{n}(-\sin(\frac{1}{\sqrt{n}})$
28. $\displaystyle \lim_{x\to-\infty}(\sin x+2)\ln(-x)$
29. $\displaystyle \lim_{x\to\infty}\sqrt[x]{1+\sin(x)}$
30. $\displaystyle \lim _{n\rightarrow \infty }\dfrac {1+\cos \dfrac {x} {n}+\cos \dfrac {2x} {n}+\ldots +\cos\dfrac {\left( n-1\right) x} {n}} {n } = \dfrac{\sin x}{x}$
31. $\displaystyle \lim_{n\to\infty} \sin\bigl(\pi\sqrt{n^2+1}\bigr)=0 $
32. $\displaystyle \lim_{x\to\infty} \frac {7 \sin x}{\sqrt{5x}}$
33. $\displaystyle \lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$
34. $\displaystyle \lim_{x\to\infty}\frac{\sin^2x}{x^2}$
35. $\displaystyle \lim_{n\to \infty}\frac{\sin n}{n}=0$
36. $\displaystyle \lim_{x\to \infty} 5^x \sin\left(\frac{a}{5^x}\right)$
37. $\displaystyle \lim_{x\to\infty} \frac{x-\sin x}{x+\cos x} = \lim_{x\to\infty} \frac{1-\cos x}{1-\sin x} $
38. $\displaystyle \lim \limits_{n \to \infty} \sqrt[n]{4n + \sin \sqrt{n} + \cos (\tfrac{1}{n^2}) + 17}$
39. $\displaystyle \lim_{x\rightarrow +\infty}\frac{x(1+ \sin(x))}{x-\sqrt{1+x^2}}$
40. $\displaystyle \lim_{x\to\infty}\frac{x-\frac{1}{2}\sin x}{x+\frac{1}{2}\sin x} $
41. $\displaystyle \displaystyle \lim_{x\to \infty} \dfrac{x + 5 \sin x}{x-\cos x} = 1$
42. $\displaystyle \lim \limits_{n\to\infty} \frac{n}\pi \cos \left( \frac{2\pi}{3n}\right) \sin \left( \frac{4\pi}{3n}\right)$
43. $\displaystyle \lim_{x\to\infty}\frac{\sin (x)+\cos (3x)}{x+2}\;\;?$
44. $\displaystyle \lim_{n\to\infty}\sin\left(\frac{1}{n}\right)=\sin\left(\lim_{n\to\infty}\frac{1}{n}\right)=\sin(0)=0$
45. $\displaystyle \lim_{n\rightarrow \infty} \sin((2n\pi + \frac{1}{2n\pi}) \sin(2n\pi + \frac{1}{2n\pi}))$
46. $\displaystyle \lim_{x\to\infty}\frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$
47. $\displaystyle \lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$
48. $\displaystyle \lim_{x \to \pm \infty}\left(\cos\left(e^{x^{1/3}+\sin x}\right)\right) $
49. $\displaystyle \lim_{x\rightarrow \infty} x\sin \frac {c}{x} $
50. $\displaystyle \lim_{ x\to \infty}\cos x^{\frac{1}{\sin x}}$
51. $\displaystyle \lim_{n\rightarrow \infty} n\sin(2\pi e n!)$
52. $\displaystyle \lim_{n \to \infty} \frac{(4n^3 + 1)(4n - 2)!n\sin{\frac{2}{n}}}{(4n + 1)!+3} $
53. $\displaystyle \lim_{x\to +\infty}\dfrac {\cos^5x\sin^5x} {x^8\sin^2x-2x^7\sin x\cos^2x+x^6\cos^4x+x^2\cos^8x}$
54. $\displaystyle \lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$
55. $\displaystyle \lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$
56. $\displaystyle \lim_{n \to \infty} \frac{5 n^2 +\sin n}{3 (n+2)^2 \cos(\frac{n \pi}{5})},$
57. $\displaystyle \lim_{x \to \infty}\frac{x-\sin x }{x-\tan x}$
58. $\displaystyle \lim_{x\to\infty}\left(\sin\frac1x+\cos\frac1x\right)^x=e.$
59. $\displaystyle \lim_{x \to \infty}{x^{\frac{5}{3}}\cdot\left[{\left(x+\sin{\frac{1}{x}}\right)}^{\frac{1}{3}} -x^{\frac{1}{3}}\right]}$
60. $\displaystyle \mathop{\lim}\limits_{x \to \infty}x\left(\frac{1}{x}\sin x-1+\frac{1}{x}\right)=(0k-1+0)\cdot\mathop{\lim}\limits_{x \to \infty}x,$
61. $\displaystyle \lim\limits_{n\to\infty}\sin n$
62. $\displaystyle \lim_{x\rightarrow\infty}\;\sin(x)?$
63. $\displaystyle \lim_{n\to\infty}\sqrt[n]{\frac{|\sin1|}1+\cdots+\frac{|\sin n|}{n}\ }\,.$
64. $\displaystyle \displaystyle\lim_{n\to\infty}|\sin n|^{\frac{1}{n}}$
65. $\displaystyle \lim_{n\rightarrow\infty} n^{\sin(\pi/n)}$
66. $\displaystyle \lim_{x \to +\infty} \frac{x - \sin(x) \log(1+x)}{x^7}$
67. $\displaystyle \lim_{n \to \infty } \left |\sin n \right |n=\infty $
68. $\displaystyle \displaystyle \lim_{n\rightarrow \infty}\frac{\ln (1+n^{3})-\ln(n^{6})}{\sin ^{3}(n)} $
69. $\displaystyle \lim _{n\rightarrow \infty }\dfrac {1} {n}\left( \dfrac {\sin \dfrac {x} {2}\cos \dfrac {x} {2}\left(\dfrac {1} {n}-1\right)} {\sin \dfrac {x} {2n}}\right) $
70. $\displaystyle \lim_{n\to \infty} \sin{1\over n}=0$
71. $\displaystyle \displaystyle \limsup_{n\to \infty} \sin(n) = 1$
72. $\displaystyle \lim_{n\to\infty}n^2\left(n\sin{(2e\pi\cdot n!)}-2\pi\right)=\dfrac{2\pi(2\pi^2-3)}{3}$
73. $\displaystyle \lim_{n\to\infty}(\sin\frac{\ln2}{2}+\sin\frac{\ln3}{3}+\ldots+\sin\frac{\ln n}{n})^{\frac{1}{n}}=1$
74. $\displaystyle \lim_{n \rightarrow \infty} n \sin(2 \pi e n!)$
75. $\displaystyle \lim_{n\rightarrow\infty}X_n \sin\left(\frac{1}{X_n}\right)=0$
76. $\displaystyle \lim\limits_{x \to \infty}\dfrac{\sin(5x)}{\sin(x)\sin(2x)}$
77. $\displaystyle \lim\limits_{n\to\infty}(\sin(n)-n)=-\infty$
78. $\displaystyle \lim_{n\to\infty} \dfrac{\ 1+|\sin n|}{2n}$
79. $\displaystyle \lim_{x \rightarrow \infty} x\sin (1/x) $
80. $\displaystyle \lim_{n\to\infty} \frac{\cot{\frac{2}{n}}+n\csc{\frac{3}{n^3}}}{\csc{\frac{3}{n}} + n\cot{\frac{2}{n^2}}} = \lim_{n\to\infty} \frac{\frac{\frac{2}{n}}{\tan{\frac{2}{n}}}\cdot\frac{1}{2n^2}+\frac{\frac{3}{n^2}}{\sin{\frac{3}{n^2}}}\cdot\frac{1}{3}}{\frac{\frac{3}{n}}{\sin{\frac{3}{n}}}\cdot \frac{1}{3n^2}+\frac{\frac{2}{n^2}}{\tan{\frac{2}{n^2}}}\cdot\frac{1}{2}}=...=\frac{2}{3}$
81. $\displaystyle \lim_{x\rightarrow \infty} x^3 \left(\tan{\frac{1}{x}}\right)\left(\sin{\frac{3}{x^2}}\right)$
82. $\displaystyle \lim_{x \to \infty} \sqrt{\sin{\frac{3}{\sqrt{x}}}} = \sqrt{\sin{3 \sqrt{ \lim_{x \to \infty} \frac{1}{x} }}}$
83. $\displaystyle \lim_{x\to\infty} x\sin({1\over x})$
84. $\displaystyle \lim_{n \rightarrow \infty} \frac{6n^3+2n^2-7}{(n+\sin(n^2))(n^2+1)} =6$
85. $\displaystyle \lim_{n\to\infty} \sin\frac{2\pi}{3^{n}} \sim \frac{2\pi}{3^{n}}\tag{3}$
86. $\displaystyle \lim_{x\to\infty}\dfrac{4x^3 - 2x + 1}{8x^3 + \sin(x^2) - x^{-1}}$
87. $\displaystyle \lim_{n \to \infty} (\sin(x))^n $
88. $\displaystyle \lim_{x \to \infty} {\sin(x-\lfloor x\rfloor)}$
89. $\displaystyle \lim_{n \to +\infty}2^n\sin(2^{-n}) = 1$
90. $\displaystyle \lim_{n\to\infty}n\sin\left(\frac1n\right)=1?$
91. $\displaystyle \displaystyle \lim \limits_{x \to \infty} \frac{3 - \sin(e^x)}{\sqrt{x^2 + 2}}$
92. $\displaystyle \lim_{n\to\infty}\frac{3^n+2n^n+n!}{(n+1)^4+\sin n+(3n)!}$
93. $\displaystyle \lim_{n\to\infty}\left|\sin n\right|^\frac1n$
94. $\displaystyle \lim_{n \rightarrow \infty} n \sin (2\pi e n!).$
95. $\displaystyle \lim_{n\to\infty+}\frac{\frac{\cos(n!)}{n!+2n}-\sin\left(\frac1{n^2}\right)(n^8+\ln n)^{\frac14}}{(n^3+n^2)^{\frac13}-n}$
96. $\displaystyle \lim_{n\to\infty}\underbrace{\sin{\sin{\cdots\sin{x}}}}_{n},x\in R$
97. $\displaystyle \lim_{x\to \infty} x^2 \sin\dfrac{1}{x^2}$
98. $\displaystyle \lim_{x\to\infty}\frac{\sin[xf(x)]}{x\cdot\sin[f(x)]}=1$
99. $\displaystyle \lim\limits_{x \to \infty} f(x)=\lim_{x\rightarrow \infty}\frac{x+\sin(x)}{x+1}=1.$

Answer 5

Trigonometric limits: with cosine (no sine), at $0$

1. $\displaystyle \lim_{x\to0}\frac{1-\cos(x)}{x} $ 1, 2, 3, 4
2. $\displaystyle \lim\limits_{x \to 0^+} \frac{\ln[\cos(x)]}{x}$
3. $\displaystyle \displaystyle\lim_{x\to 0}\frac{\frac{1}{2}x^{-1/2}}{\frac{1}{2}\frac{1}{\sqrt{x}}+\frac{1}{2}\frac{1}{\sqrt{x}}\cos\sqrt{x}}$
4. $\displaystyle \displaystyle\lim_{x \to 0}\frac{1-\cos (1- \cos x)}{x^4}$
5. $\displaystyle \lim_{x\to 0} \frac{\cos(x)}{x}$
6. $\displaystyle \lim\limits_{x\to 0} \frac{\ln (\cos ax)}{\ln (\cos bx)}$
7. $\displaystyle \lim_{x\to 0} {1-\cos x\over x^2} = \frac12$ 1, 2, 3
8. $\displaystyle \lim_{x\rightarrow 0} \cos(\frac{1}{x})$
9. $\displaystyle \lim_{x\to 0}\frac{1-\cos 2x}{1-\cos x}$
10. $\displaystyle \lim_{x \to 0} \frac{1- \cos 5x}{x^2} = \frac{25}{2}.$
11. $\displaystyle \lim_{x\to 0} \frac{1-\cos(3x)}{2x^2}$
12. $\displaystyle \lim_{x\to 0}\frac{\cos 3x-\cos x}{x^2}$
13. $\displaystyle \lim_{x\to0} \frac{\cosh x\cosh 2x\cosh 3x \cdots \cosh nx-1}{x^2}$
14. $\displaystyle \lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x} $
15. $\displaystyle \lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$
16. $\displaystyle \lim_{(a,b)\to (0,0)} \frac{\cos^{-1}(f(a,b,\lambda,\gamma))}{\cos^{-1}(f(a,b,1,\gamma))}=|\lambda|$
17. $\displaystyle \lim_{x \to 0} (\frac{\cos x } {x e^{x}}- \frac{1}{x})$
18. $\displaystyle \lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)$
19. $\displaystyle \lim\limits_{(x,y) \to (0,0)} f(x,y) = \dfrac{\cos(x) -1 - {x^2 \over 2}}{x^4 + y^4}$
20. $\displaystyle \lim_{x\rightarrow 0}\left[ \frac{\ln(\cos x)}{x\sqrt{1+x}-x} \right]$
21. $\displaystyle \lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \lim_{x\rightarrow 0}\frac{6\cos^3(2x)}{48\cos^3(2x)} = \frac{6}{48} = 0.125$
22. $\displaystyle \lim_{x\rightarrow 0}\frac{\arcsin x - \arctan x}{e^x-\cos x -x^2 -x}$
23. $\displaystyle \lim_{x\to 0}\frac{\tan x(1- \cos x)}{3x^2} $
24. $\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{\cos {(xy)} - 1}{x^2y^2}$
25. $\displaystyle \lim_{x \to 0} x^2 \cos\left(\frac{1}{x^2}\right) = 0$
26. $\displaystyle \lim_{(x,y)\to (4,0)}y^2\cos\left({1\over(x-4)^2+y^2}\right) $
27. $\displaystyle \lim\limits_{x \to 0}{\frac{1-\cos(1-\cos x)}{x^4}}$
28. $\displaystyle \lim_{x \rightarrow 0} \, x^{1/3}\cos(1/x) = 0.$
29. $\displaystyle \lim_{t \to 0}\frac{t^2}{1-\cos^2t} $
30. $\displaystyle \lim_{x\to 0}\frac{(1-3x)^\frac{1}{3} -(1-2x)^\frac{1}{2}}{1-\cos(\pi x)}. $
31. $\displaystyle \lim_{x \to 0} \frac{x \cdot \operatorname{cosec}(2x)}{\cos(5x)}$
32. $\displaystyle \lim_{x \to 0}\frac{\left [ \cos(a+x)-\cos(a-x) \right ]^2}{\tan^2(3x)} $
33. $\displaystyle \lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$ 1, 2
34. $\displaystyle \displaystyle\lim_{x\to 0 }\frac{e^{\arctan(x)}-e^{\arcsin(x)}}{1-\cos^3(x)}$
35. $\displaystyle \displaystyle\lim_{x \to 0}\frac{1-\cos(ax)}{ax}=0$
36. $\displaystyle \lim_{x\to 0} \frac{\ln(\cos x)}{\ln(\cos 3x)}$
37. $\displaystyle \lim_{h\to 0}h\cos\frac1h\stackrel{?}=0$
38. $\displaystyle \lim_{x \rightarrow 0}\left(\frac{\tan \left(\pi\cos^{2}x\right)}{x^2} \right)$
39. $\displaystyle \lim_{x\to 0}\dfrac{\sqrt[m]{\cos x}-\sqrt[n]{\cos x}}{x^2}$
40. $\displaystyle \lim_{x\to 0}\frac1x\cos(\frac1x)$
41. $\displaystyle \lim_{h\rightarrow 0} \frac{|\cos h-1|}{h}$
42. $\displaystyle \lim_{x\to0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}}$
43. $\displaystyle \lim_{h \to 0} \frac {\arccos(\cos^2h)} {h}$
44. $\displaystyle \lim_{x \to 0} \frac{\cos3x-\cos x}{\tan2x^2}$
45. $\displaystyle \lim_{x\to0}\cos\left(\frac{1}{x}\right)$
46. $\displaystyle \lim_{x \to 0} \frac{2 - \cos(3x) - \cos(4x)}{x}?$
47. $\displaystyle \lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$
48. $\displaystyle \lim_{x\to 0^-} \frac{2+p\cos{x}}{3-2\cos{x}} = 0$
49. $\displaystyle \lim\limits_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2+y^2}$
50. $\displaystyle \displaystyle \lim_{k \to 0}{f(k) = 2} \;+\; \lim_{k \to 0}{k^{\frac{3}{2}}\cos {\frac{1}{k^2}}}$
51. $\displaystyle \lim_{x\to 0} \frac{e^{2x}-x^2+x}{\cos(x)-1} = ~? $
52. $\displaystyle \lim_{x\to 0} \frac{1-\cos x \cdot \sqrt{\cos2x} }{x^2}$
53. $\displaystyle \lim_{h \rightarrow 0}\frac{\left| \cos(\frac{\pi}{2+ \pi h}) \right| - \left|cos(\frac{\pi}{2})\right|}{h}$
54. $\displaystyle \lim_{x\to 0}\frac{1}{x}\cdot\left[\arccos\left(\frac{1}{x\sqrt{x^{2}- 2x\cdot \cos(y)+1}}-\frac{1}{x}\right)-y\right] $
55. $\displaystyle \lim_{h \rightarrow 0}{ \left|{\cos h \over h}\right|}$
56. $\displaystyle \lim_{r\to 0}\frac{2\cos \theta}{r^2+\cos\theta}=\frac{2\cos \theta}{\cos\theta}=2$
57. $\displaystyle \lim_{h\to0} \frac{\cos(\frac{\pi}{3}+h)-\frac{1}{2}}{h}$
58. $\displaystyle \lim_{x\to0} \frac{2^x-\cos\left(x\right)}{3^x-\operatorname{ch}\left(x\right)}$
59. $\displaystyle \lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})}$
60. $\displaystyle \displaystyle \lim_{x\to 0} \cos x=1$
61. $\displaystyle \lim_{x\to 0} \frac{1-\cos x \cos(2x)}{x^2} \, . $
62. $\displaystyle \displaystyle\lim _{x\to \:0}\left(\dfrac{\sqrt{1-\cos \left(2x\right)}}{\sqrt{2}x}\right)$
63. $\displaystyle \lim_{x \to 0} {x^2 \over \cos (3x) - 1}$
64. $\displaystyle \lim_{h \to 0}\cos(a + h) = \cos(a)$
65. $\displaystyle \lim_{x\to 0} \left(\csc(x^2)\cos(x)-\csc(x^2)\cos(3x) \right)$
66. $\displaystyle \lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$
67. $\displaystyle \lim_{x\to 0} \frac {1-\cos x}{1-\sec^2x}$
68. $\displaystyle \lim_{x \rightarrow 0} \frac{1 - \cos{x}}{x}=0$
69. $\displaystyle \lim_{x \to 0} |x|^\alpha\cos(1/x^2) = 0.$
70. $\displaystyle \ \lim_{ x \to 0}\frac{\cos(x)}{x} $
71. $\displaystyle \lim_{r \rightarrow 0} \frac{2r\cos(r^2)}{2r} $
72. $\displaystyle \lim_{x\to 0} x^2\cos\left(\frac1x\right)$
73. $\displaystyle \displaystyle\lim_{v\to180}\frac{360\cos\left(\dfrac{v}{2}\right)}{180-v}$
74. $\displaystyle \lim\limits_{x\to 0}\left(\frac{1-\cos(2x)}{1-\cos(4x)}\right)^2 $
75. $\displaystyle \lim_{x \to 0}\frac{x}{\sqrt{1-\cos2x}}$
76. $\displaystyle \lim_{y \to 0^+} (\cosh (3/y))^y$
77. $\displaystyle \lim\limits_{x\to 0}\frac{({\ln(1+x) -x +\frac{x^2}{2})^4}}{(\cos(x)-1+\frac{x^2}{2})^3}$
78. $\displaystyle \displaystyle\lim_{z\to0} \operatorname{pv}\left(\cos(z)^\frac{1}{z^2}\right)$
79. $\displaystyle \lim_{x \to 0}(\cos x)^{1/x^4}$
80. $\displaystyle \lim_{x\to 0} \, \cos \left(\frac{\pi -\pi \cos ^2(x)}{x^2}\right)$
81. $\displaystyle \lim_{x \to 0}\frac{1-\cos(3x)}{2x^2}$
82. $\displaystyle \lim_{x \to 0}\frac{x \csc 10x}{\cos20x}$
83. $\displaystyle \lim_{x \to 0}\frac{(e^{2\tan(x)}-1) \cdot \ln(2-\cos^2(x))}{\sqrt{1+x^3}-(\cos x)}$
84. $\displaystyle \lim_{x \rightarrow 0}\left(\frac{(\cos{x}-1)(\cos{x}-e^{x})}{x^{n}} \right)$
85. $\displaystyle \lim_{x\to 0}\frac{\ln(\cosh(x))}{x\ln(1+x)}$