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Math Help - Limit with Taylor formula

  1. #1
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    Limit with Taylor formula

    "Calculate the limit using Taylor's formula". I know that the limit should be 1/30. Problem is how to prove it.

    <br />
\lim_{x\rightarrow0}\frac{tan(sinx) - sin(tanx)}{x^7}<br />
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by xkyve View Post
    "Calculate the limit using Taylor's formula". I know that the limit should be 1/30. Problem is how to prove it.

    <br />
\lim_{x\rightarrow0}\frac{tan(sinx) - sin(tanx)}{x^7}<br />
    hint:

    \sin x = x - \frac {x^3}{3!} + \frac {x^5}{5!} - + \cdots for all x

    and

    \tan x = x + \frac {x^3}{3} + \frac {2x^5}{15} + \cdots for |x| < \frac {\pi}2
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  3. #3
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    i know the expansions and i found the value 17/315 + 1/7! for the limit but it's incorrect. i think i must find a path to write those trig. functions in another way.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Let f(x)=\tan(\sin(x)) and g(x)=\sin(\tan(x)). Then

    f(0)=0 ---- g(0)=0
    f'(0)=1 ---- g'(0)=1
    f''(0)=0 ---- g''(0)=0
    f^{(3)}(0)=1 ---- g^{(3)}(0)=1
    f^{(4)}(0)=0---- g^{(4)}(0)=0
    f^{(5)}(0)=-3 ---- g^{(5)}(0)=-3
    f^{(6)}(0)=0 ---- g^{(6)}(0)=0
    f^{(7)}(0)=-107---- g^{(7)}(0)=-275


    \therefore\quad\tan(\sin(x))=x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{107x^7}{5040}\pm\cdots

    and \sin(\tan(x))=x+\frac{x^3}{6}-\frac{x^5}{4}-\frac{275x^7}{5040}\pm\cdots


    \begin{aligned}\therefore\quad&\lim_{x\to{0}}\frac  {\tan(\sin(x))-\sin(\tan(x))}{x^7}\\<br />
&=\lim_{x\to{0}}\frac{x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{107x^7}{5040}\pm\cdots-\left(x+\frac{x^3}{6}-\frac{x^5}{4}-\frac{275x^7}{5040}\pm\cdots\right)}{x^7}\\<br />
&=\lim_{x\to{0}}\frac{\frac{x^7}{30}\pm\cdots}{x^7  }\\<br />
&=\frac{1}{30}<br />
\end{aligned}
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  5. #5
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    thank you
    one more question, did u do those computations by hand?
    Last edited by xkyve; November 10th 2008 at 01:50 AM.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by xkyve View Post
    thank you
    one more question, did u do those computations by hand?
    Up to the third derivative. And then I was like screw this and used my calculator.
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  7. #7
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    yeah, those computations are long... anyways, i was wondering if there was another method, one that envolves only a pencil and some paper. i'll have to think about it...
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by xkyve View Post
    yeah, those computations are long... anyways, i was wondering if there was another method, one that envolves only a pencil and some paper. i'll have to think about it...
    There are other methods, but none that are using the Taylor method. Because that is the Taylor method.
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  9. #9
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    probably you are right, but i will still try to solve it using taylor but less computations.
    doesn't matter, can u recommend me some good software that is able to compute derivatives, limits, and so on?

    thanks
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  10. #10
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by xkyve View Post
    probably you are right, but i will still try to solve it using taylor but less computations.
    doesn't matter, can u recommend me some good software that is able to compute derivatives, limits, and so on?

    thanks
    Mathcad, TI-89, or Mathematica are all good choices.
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