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anonymous

  • one year ago

Another limit question...

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  1. anonymous
    • one year ago
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    \[\lim_{n \rightarrow \infty}\ln\left( 1+ \frac{ 4-\sin(x) }{ n } \right)^n\]

  2. anonymous
    • one year ago
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    My approach was that I raised it to the power of e and then I Just found the limit of the stuff inside. But that's as far as I got.... I see a pattern of (1+(1/x))^n which is the definition of e but I don't really know about this one...

  3. hartnn
    • one year ago
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    4 -sin x or 4 - sin n ?? if its 4- sin x, then its a constant...

  4. anonymous
    • one year ago
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    Nope that is not a typo. It is indeed a constant.

  5. hartnn
    • one year ago
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    cool! so you have the function of the form \((1+ax)^{1/x}\) right?

  6. anonymous
    • one year ago
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    How so?

  7. anonymous
    • one year ago
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    Okay well I can see it if you make a substitution.

  8. anonymous
    • one year ago
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    Go on.

  9. hartnn
    • one year ago
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    good, i'll tell you what we do in case of \((1+ax)^{1/x}\) -- > \((1+ax)^{1/x} = [(1+ax)^{\frac{1}{ax}}]^a\) and then use the limit formula, (if you can use) \(\lim \limits_{x\to \infty} (1+1/x)^x = e\)

  10. anonymous
    • one year ago
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    Brilliant! But the limit would then be 1 not 0...

  11. anonymous
    • one year ago
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    Which according to wolfram it's 0.

  12. anonymous
    • one year ago
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    http://www.wolframalpha.com/input/?i=lim+n-%3E+infinity+ln%281%2B%284-sin%28x%29%29%2Fn%29%5En

  13. anonymous
    • one year ago
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    Ohh wait. I goofed. Have to take the ln of that.

  14. anonymous
    • one year ago
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    ln(1) is 0. Thank you!

  15. hartnn
    • one year ago
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    in the wolf, its shows ln^n

  16. anonymous
    • one year ago
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    Yeah it's a notational thing. That just the inside raised to the n.

  17. anonymous
    • one year ago
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    Kinda like sin^2(x) and (sin(x))^2.

  18. hartnn
    • one year ago
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    okk... i thought the answer would be 4- sin x ..

  19. anonymous
    • one year ago
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    Nah. It's 0.

  20. hartnn
    • one year ago
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    :)

  21. anonymous
    • one year ago
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    @hartnn : So I got up here.

  22. anonymous
    • one year ago
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    \[\lim_{b \rightarrow 0}(1+(4-\sin(x)b)^{\frac{ 1 }{ b }}\]

  23. anonymous
    • one year ago
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    Is that okay so far or am I way off?

  24. hartnn
    • one year ago
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    i assume there is ln outside of that limit and you just plugged in b =1/n

  25. anonymous
    • one year ago
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    Indeed sir.

  26. hartnn
    • one year ago
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    yes, go on

  27. anonymous
    • one year ago
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    Stuck >.< .

  28. anonymous
    • one year ago
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    Like I know that should be e but I'm having trouble relating it to the definition.

  29. hartnn
    • one year ago
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    whatever expression is with \(\Large 1+ ...\) that same expression should be with \(\Large \dfrac{1}{...}\) thats how I remember so we have 1+ (4-sin x)b so the fraction in the exponent should be \(\Large \dfrac{1}{(4-\sin x)b}\)

  30. hartnn
    • one year ago
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    |dw:1443855809053:dw|

  31. anonymous
    • one year ago
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    But the definition of e is (1+1/x)^x right? Here we have (1+(constant)b)^(1/b) . Are those equivalent?

  32. hartnn
    • one year ago
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    http://www.wolframalpha.com/input/?i=lim+x-%3E+infty+%281%2Bax%29%5E%281%2F%28ax%29%29+

  33. hartnn
    • one year ago
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    that will clear your doubt

  34. anonymous
    • one year ago
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    Interesting. I did not know that even after 4 years of calculus and differential equations lol. I learn new things every day!

  35. Jhannybean
    • one year ago
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    I want to learn how to solve this as well.. I get some steps but Im confused on others. :(

  36. hartnn
    • one year ago
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    We take that as a formula, but its easy to prove that using L'Hopital's rule.

  37. hartnn
    • one year ago
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    I will be writing out all the steps from the beginning

  38. anonymous
    • one year ago
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    Yes I know we can use L'hopital's rule but for this assignment they (Other students) can't use that.

  39. hartnn
    • one year ago
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    *drawing Let 4 - sin x = a , since its a constant. |dw:1443856327983:dw|

  40. anonymous
    • one year ago
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    Yep I got that.

  41. hartnn
    • one year ago
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    writing these steps for everyone's benefit :) |dw:1443856406335:dw|

  42. anonymous
    • one year ago
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    Yep makes sense so far...

  43. hartnn
    • one year ago
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    |dw:1443856506563:dw|

  44. hartnn
    • one year ago
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    that big bracket evaluates to 1 1^a = 1 ln 1 = 0 :D

  45. anonymous
    • one year ago
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    Cool!

  46. anonymous
    • one year ago
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    Is there somewhere I can find the proof of the stuff in the brackets?

  47. Jhannybean
    • one year ago
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    |dw:1443856722618:dw|

  48. anonymous
    • one year ago
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    Yeah to get into the proper form for the limit.

  49. hartnn
    • one year ago
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    lets prove it :) I'll use L'Hopitals, need to search the net for other proofs. |dw:1443856725213:dw| quick check, 0/0 form ln (1+ab) = ln 1 = 0 ab = 0 so we can apply L'Hopital's rule here

  50. anonymous
    • one year ago
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    Nice!

  51. hartnn
    • one year ago
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    we can do all kinds of mathematically legal manipulations to bring an expression in the standard form. i needed a form like (1+x)^(1/x) thats why I multiplied and divided by 'a' , which should be NON-ZERO (point to be noted.)

  52. Jhannybean
    • one year ago
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    oh I see I see

  53. Astrophysics
    • one year ago
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    How does this 0, the power would be undefined |dw:1443857378987:dw|

  54. Astrophysics
    • one year ago
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    Oooh wait nvm, n = 1/b when n-> infinity, b ->9

  55. Astrophysics
    • one year ago
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    b->0*

  56. Jhannybean
    • one year ago
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    Yeah, there you go

  57. Astrophysics
    • one year ago
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    Haha, I totally missed that, ok it's good now. Great explanation @hartnn thanks

  58. Jhannybean
    • one year ago
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    me 4.

  59. anonymous
    • one year ago
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    @hartnn

  60. anonymous
    • one year ago
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    |dw:1443858029848:dw|

  61. hartnn
    • one year ago
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    http://www.wolframalpha.com/input/?i=lim+n-%3E+infty+%281%2Ba%2Fn%29%5E%28%28n%29%29+ ln e^a = a ln e = a a = 4-sin x thats what I first got. but this wolf answer got me all confused and I ended up using b->infty instead of b->0

  62. anonymous
    • one year ago
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    But that's wrong though. b goes to 0, not n.

  63. anonymous
    • one year ago
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    not infinity*

  64. Jhannybean
    • one year ago
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    can you explain as to `why` b \(\rightarrow\) 0 and not b \(\rightarrow \infty\) ?

  65. hartnn
    • one year ago
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    true, and it makes sense logically too, constant/n = very very small no. 1+ very very small no. = 1 1^ very very larger number = 1 ln 1 =0 so the limit must go to 0 but with all the mathematical steps, I still get the answer as 4-sin x

  66. anonymous
    • one year ago
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    It's 0 according to wolf :( .

  67. anonymous
    • one year ago
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    Asking around. Like my feel is that the inside of that logarithm should be a 1.

  68. anonymous
    • one year ago
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    Only then can we get a 0.

  69. hartnn
    • one year ago
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    inside of a logarithm http://www.wolframalpha.com/input/?i=lim+n-%3E+infty+%281%2B%284-sin+x%29%2Fn%29%5E%28%28n%29%29+

  70. anonymous
    • one year ago
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    It's witchcraft I tell you!

  71. anonymous
    • one year ago
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    This is the statement of t he original problem.

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  72. hartnn
    • one year ago
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    we can only bring limit inside a function if that function is continuous. and logarithm is indeed continuous...

  73. hartnn
    • one year ago
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    http://www.wolframalpha.com/input/?i=lim+n-%3E+infinity+n*+ln+%281%2B%284-sin%28x%29%29%2Fn%29 even that gives 4-sin x!

  74. Jhannybean
    • one year ago
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    haha oh my goodness x_x

  75. anonymous
    • one year ago
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    Ohh wow. Wolfram is apparantly wrong.

  76. Jhannybean
    • one year ago
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    LOL

  77. Jhannybean
    • one year ago
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    "which is apparently related to e" xD

  78. Jhannybean
    • one year ago
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    http://i.imgur.com/F4xr6yp.png

  79. anonymous
    • one year ago
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    Okay wow wolfram can't read notation clearly -.- .

  80. hartnn
    • one year ago
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    its very rare case where wolfram goes wrong

  81. anonymous
    • one year ago
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    Wow ._. ...

  82. hartnn
    • one year ago
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    \(\color{blue}{\text{Originally Posted by}}\) @hartnn okk... i thought the answer would be 4- sin x .. \(\color{blue}{\text{End of Quote}}\) and then wasted an hour :P

  83. anonymous
    • one year ago
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    All because we hail our god wolfram alpha too much ._. ...

  84. Jhannybean
    • one year ago
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    xD It hardly fails!!! As humans we value consistency and dependability :P

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