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Dido525

  • 3 years ago

Find the values of a and b so that the following is true.

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  1. Dido525
    • 3 years ago
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    I am thinking about L'Hospital's rule but I am not sure how I would apply it...

  2. Rezz5
    • 3 years ago
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    well, you can separate them, then solve them

  3. Dido525
    • 3 years ago
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    Yeah.... But I have two variables then.

  4. Rezz5
    • 3 years ago
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    are you sure it tends to infinity?

  5. Dido525
    • 3 years ago
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    Yep.

  6. Dido525
    • 3 years ago
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    0.

  7. Dido525
    • 3 years ago
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    oops.

  8. Rezz5
    • 3 years ago
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    why two variables?

  9. Rezz5
    • 3 years ago
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    if you differentiate the variable you get nothing from a, but b will still be present, you can get b right?

  10. Dido525
    • 3 years ago
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    \[\lim_{x \rightarrow 0} (\frac{ \sin(2x) }{ x^3 }+a+\frac{ b }{ x^2 })=0\]

  11. Dido525
    • 3 years ago
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    Yeah. I think you are right.

  12. Rezz5
    • 3 years ago
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    ahhh, so it tends to zero?

  13. Dido525
    • 3 years ago
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    Yep :P . I typed wrong.

  14. Rezz5
    • 3 years ago
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    no problem, you can separate them then do the limits, you can spot it by inspection, very easily

  15. Dido525
    • 3 years ago
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    Hmm Let me try...

  16. Rezz5
    • 3 years ago
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    have you done it?

  17. Dido525
    • 3 years ago
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    I can't solve the limit of: \[\lim_{x \rightarrow 0}\frac{ \sin(2x) }{ x^3 }\]

  18. Rezz5
    • 3 years ago
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    you can!! differentiate 3 times until the denominator is 1,

  19. Rezz5
    • 3 years ago
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    not 1, differentiate until the denominator is a constant

  20. Rezz5
    • 3 years ago
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    sin(2)=2sin(x)cos(x)

  21. Dido525
    • 3 years ago
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    I got -4/3 .

  22. Rezz5
    • 3 years ago
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    for?

  23. Dido525
    • 3 years ago
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    The limit.

  24. Rezz5
    • 3 years ago
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    sin(2x)/x^3 as x tends to infinity?

  25. Rezz5
    • 3 years ago
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    how ?

  26. Dido525
    • 3 years ago
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    To 0 :P .

  27. Rezz5
    • 3 years ago
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    lol!!! im losing it today...

  28. Rezz5
    • 3 years ago
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    but still, i would have thought it was zero

  29. Dido525
    • 3 years ago
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    Waitt.

  30. Dido525
    • 3 years ago
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    The Limit is infinity..... :( .

  31. Rezz5
    • 3 years ago
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    yep,

  32. Dido525
    • 3 years ago
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    We can't solve anything then.

  33. Rezz5
    • 3 years ago
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    we could, i mean if one of the variables was infinity

  34. Dido525
    • 3 years ago
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    But We can't do anything with infinity.... It's not a number.

  35. Rezz5
    • 3 years ago
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    wait are you sure it should tend to 0?

  36. Dido525
    • 3 years ago
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    Well I would assume it should tend to a finite value.

  37. Rezz5
    • 3 years ago
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    i suspect the limit should tend to infinity then for it so satisfy the limit

  38. Rezz5
    • 3 years ago
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    not necessarily, it can be infinity, let assume it was infinity, then we can do the question with ease. have you done analysis by any chance?

  39. Dido525
    • 3 years ago
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    No.....

  40. Dido525
    • 3 years ago
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    Here is the question.

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  41. Rezz5
    • 3 years ago
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    okay, no problem, assume it is infinity because i think you were right the first time round, if it was infinity just plug in infinity to the differential you had computed

  42. Dido525
    • 3 years ago
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    If we did that then a would be infinity and b would not exist.

  43. Rezz5
    • 3 years ago
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    b would, it would be x^2,

  44. Rezz5
    • 3 years ago
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    one moment,

  45. Dido525
    • 3 years ago
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    but since b^2 tends to infinity the whole thing would be 0.

  46. Dido525
    • 3 years ago
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    x^2 sorry not b^2.

  47. Rezz5
    • 3 years ago
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    which would be 0, at when b/x^2 as x ->0

  48. Dido525
    • 3 years ago
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    Exactly. We can't work with that.

  49. Rezz5
    • 3 years ago
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    we could, if b = 0,

  50. Rezz5
    • 3 years ago
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    0/anynumber = 0

  51. Dido525
    • 3 years ago
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    But it's not. B/x^2 is zero, not b.

  52. Dido525
    • 3 years ago
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    We don't know what b is.

  53. Rezz5
    • 3 years ago
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    a could be -infinity, then this would satisfy the limit

  54. Dido525
    • 3 years ago
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    But infinity- infinity is not 0.

  55. Rezz5
    • 3 years ago
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    yes it is

  56. Dido525
    • 3 years ago
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    No it's not. Infinity is not a number. It's a concept.

  57. Rezz5
    • 3 years ago
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    that is true, i moment, i am trying to multi task, i jst remembered the proof that it isnt,

  58. Dido525
    • 3 years ago
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    http://en.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number

  59. Rezz5
    • 3 years ago
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    okay, you need to apply l'hpitals rule over and over again, i would recon, but first you need to put all the equation under one common factor so \[=\frac{\sin(2x) + ax^{3} + bx}{x^{3}}\]

  60. Rezz5
    • 3 years ago
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    http://answers.yahoo.com/question/index?qid=20110406112720AARpKn1 here is a better and easier way of doing it the sint is taylor expansion i think, the rest is pretty self explanatory

  61. Dido525
    • 3 years ago
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    Lol. THanks :P .

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