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WARNING: This post contains hundreds of mathematical formulas. Do not open it on devices, where rendering of MathJax is slow!

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


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

Trigonometric limits

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  • 7
    $\begingroup$ I'm scared... :-( $\endgroup$
    – Asaf Karagila Mod
    Sep 1, 2014 at 21:32
  • $\begingroup$ @AsafKaragila Main reason I went ahead with this is to see what problems will appear with such a list, and thus give more concrete input to discussion in the other thread. This is just an experiment... script time is cheap. $\endgroup$
    – user147263
    Sep 1, 2014 at 21:48
  • 1
    $\begingroup$ I don't like this! My laptop can't load more than 5 of them without freqqzin up!!! I cny even type properly.pleaseleae condense into eperate links(put all 100 in a nother linkP $\endgroup$
    – user142198
    Sep 2, 2014 at 0:15
  • $\begingroup$ @user142198 I was afraid of that... Not sure what a solution could be: moving the catalog to a site where formulas are rendered as images (would that even help?) or splitting the answers into much smaller chunks, so that they get paginated (only 30 answers are displayed at once). $\endgroup$
    – user147263
    Sep 2, 2014 at 0:17
  • $\begingroup$ If the sets were all contained in seperate links I would only need to render seta of 100 by choice, I think I m loadin 500 atm $\endgroup$
    – user142198
    Sep 2, 2014 at 0:18
  • $\begingroup$ As in have each o these answers redirect to a separate page with thelists(so yes smaller chunks), does spoiler code stop the rendering untilscrollover? $\endgroup$
    – user142198
    Sep 2, 2014 at 0:21
  • $\begingroup$ ybe these statistics arehelpful in knowing how many you want to put on the page: it takes 46sef to load first 44,84sec for 118total, 144sec for 269 otal. I have 1gb ram dual core 1.6ghz $\endgroup$
    – user142198
    Sep 2, 2014 at 0:34
  • $\begingroup$ The above numbers were amount of time elapsed since page refresh and the right was number of lines of latex had rendered thus far. Sorry if that was unclear(this was written and pasted from notepad). $\endgroup$
    – user142198
    Sep 2, 2014 at 2:06
  • $\begingroup$ +6 for your effort... $\endgroup$
    – draks ...
    Sep 2, 2014 at 6:08
  • $\begingroup$ I think it would be better if the posts about the same limit were grouped together (even if they are not exact duplicates, for example if they require some specific method, or avoiding some method). For example this limit is three times in one of your lists: math.stackexchange.com/questions/36299/…, math.stackexchange.com/questions/420698/… and math.stackexchange.com/questions/552016/… These 3 occurrences are quite far from each other. $\endgroup$ Sep 2, 2014 at 12:00
  • $\begingroup$ I suppose the ordering is by question id at the moment? It would not save that much rendering, but we could do this: $\lim\limits_{x\to0}\frac{1-\cos(x)}{x}$ 1, 2, 3 $\endgroup$ Sep 2, 2014 at 12:03
  • $\begingroup$ @MartinSleziak The order is by ad-hoc "hotness points", equal to 20*(total score of answers) plus the number of views. When the same limit appears multiple times, the first appearance should be the canonical one. Rearranging them by hand isn't in my plans; the idea was to make content better accessible with an automatic tool. $\endgroup$
    – user147263
    Sep 2, 2014 at 12:12
  • $\begingroup$ BTW some links (probably generated by some automated process) see to be incorrect. For example, one of the answers contains a link like this: $\displaystyle \lim\frac{\sin x}{x}$. It links to question about $\lim\limits_{x\to+\infty}\left(x-x^2\log\left(1+\frac{1}{x}\right)\right)$ (BTW maybe we could move the discussion to chat, so that we do not put too many comments here.) $\endgroup$ Sep 2, 2014 at 12:15
  • 1
    $\begingroup$ @MartinSleziak They are marked community-wiki. Feel free to edit, though you might find the process pretty slow. After seeing how these look and perform, I don't think I'll keep adding to this thread. Putting also logarithmic limits, exponential, square roots, rational functions... would make browsing next to impossible. I'll try to come up with a better system outside of SE. This thing can sit here as far as I'm concerned. $\endgroup$
    – user147263
    Sep 2, 2014 at 12:45
  • 1
    $\begingroup$ @MartinSleziak Don't worry about that. If I get tired of pings, I'll ask for disassociation, but so far this isn't a problem. $\endgroup$
    – user147263
    Sep 3, 2014 at 14:43

5 Answers 5

5
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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. $\displaystyle \lim_{r\rightarrow0}\frac{\cos\theta·\sin(r^2)}{r}=\lim_{r\rightarrow0}\frac{\cos\theta·r^2}{r}=\lim_{r\rightarrow0}{\cos\theta·r}=0 \text{ for all }\theta \in [0,2\pi)$
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4
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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. $\displaystyle \lim\limits_{x\to1}{\dfrac{1-x}{1-\sin\left(\dfrac{\pi \cdot x}{2}\right)}}$
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3
  • $\begingroup$ Why did you make 49 Large? $\endgroup$
    – user1729
    Sep 3, 2014 at 11:36
  • $\begingroup$ @user1729 This whole catalog was automatically generated (modulo some later edits by Martin Sleziak). The reason 49 is large is that it's large in the post. If the font should be fixed, the higher priority is fixing it there rather than here. The thing we have here is experimental. \ $\endgroup$
    – user147263
    Sep 3, 2014 at 14:41
  • $\begingroup$ Okay, that explains it. I've edited it in both places. $\endgroup$
    – user1729
    Sep 3, 2014 at 14:53
3
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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. $\displaystyle \lim_{\theta\rightarrow -\infty}\dfrac{\cos\theta}{3\theta}$
$\endgroup$
2
$\begingroup$

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. $\displaystyle \lim\limits_{x \to \infty} f(x)=\lim_{x\rightarrow \infty}\frac{x+\sin(x)}{x+1}=1.$
$\endgroup$
1
$\begingroup$

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. $\displaystyle \lim_{x\to 0}\frac{\ln(\cosh(x))}{x\ln(1+x)}$
$\endgroup$

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