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Solving limit using tailor series

  1. Mar 9, 2009 #1
    [tex]
    \lim _{x->0} \frac{cos(xe^x)-cos(xe^{-x})}{x^3}\\
    [/tex]
    [tex]
    e^x=1+x+O(x^2)\\
    [/tex]
    [tex]
    e^{-x}=1-x+O(x^2)\\
    [/tex]
    [tex]
    xe^x=x+O(x^2)\\
    [/tex]
    [tex]
    cos(x)=1-\frac{1}{2!}x^2+O(x^3)\\
    [/tex]
    [tex]
    \lim_{x->0} \frac{1-\frac{1}{2!}(x+O(x^2))^2+O(x^3)-1+\frac{1}{2!}(x+O(x^2))^2+O(x^3)}{x^3}
    [/tex]

    but i dont know how to deal with the remainders
    there squaring of them etc..

    ??
     
  2. jcsd
  3. Mar 9, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi transgalactic! :smile:

    hmm … from the x3 on the bottom, I'd guess you need to specify the x2 terms as well, and not start the Os until O(x3)

    or you could just use cosA - cosB = 2.sin(A+B)/2.sin(A-B)/2 :wink:
     
  4. Mar 9, 2009 #3
    i substituted functions one into another
    that what i got.
    where is my mathematical mistake there??

    how to open this expression and having one O()
    ??
     
  5. Mar 10, 2009 #4
    i tried to solve it again
    [tex]
    \lim _{x->0} \frac{cos(xe^x)-cos(xe^{-x})}{x^3}\\
    e^x=1+x+O(x^2)\\
    [tex]
    e^{-x}=1-x+O(x^2)\\
    [/tex]
    [tex]
    xe^x=x+x^2+O(x^2)
    [/tex]
    [tex]
    xe^{-x}=x-x^2+O(x^2)
    [/tex]
    [tex]
    cos(x)=1-\frac{1}{2!}x^2+O(x^3)\\
    [/tex]
    [tex]
    \lim_{x->0} \frac{1-\frac{1}{2!}(x+x^2+O(x^2))^2+O(x^3)-1+\frac{1}{2!}(x-x^2+O(x^2))^2+O(x^3)}{x^3}=\\
    [/tex]
    [tex]
    =\lim_{x->0} \frac{1-\frac{1}{2!}(x^2+O(x^2))+O(x^3)-1+\frac{1}{2!}(x^2+O(x^2))+O(x^3)}{x^3}=0\\
    [/tex]

    the answer is 1/2
    why i got 0??
     
  6. Mar 10, 2009 #5

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi transgalactic! :smile:

    Thta's actually pretty good, except …
    should be O(x4) at the end, not O(x3)

    and in the last line you should have x3 terms also …

    they're the ones that don't cancel! :smile:

    (and btw, it's "taylor", and why do you keep writing 2! instead of just 2? :wink:)
     
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