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Let's consider $f, g: (0, +\infty) \rightarrow\mathbb{R}$, $f(x)=\displaystyle\frac{\sin x}{x}$, $g(x)=\displaystyle\frac{\cos x}{x}$. Find the following limits

$$\lim_{n\to\infty}f^{(n)}(x)$$ $$\lim_{n\to\infty}g^{(n)}(x)$$

where $f^{(n)}$ and $g^{(n)}$ are the $n$th derivatives of $f(x)$, respectively $g(x)$.
It's a problem I thought of last days and I didn't guess the answer by trying to look at the first derivatives of both functions. What should I do here to get the limits? Thanks.

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    Do you assume this has a closed form ?2012-09-29
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    You can easily give a boundary by using taylor's theorem , but i guess you knew that already.2012-09-29
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    Maybe this is naive but if you use hypergeo form I think you can solve this.2012-09-29
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    As you take derivatives, the power of $x$ in the denominator will grow without bound.2012-09-29
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    Try exploiting $xy=\sin x$ and differentiate $n$ times using Leibniz's rule.2012-09-29
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    What makes you think that there is a well define limiting function? For example $d^n(\sin x)/dx^n$ does not converge to a limiting function as $n \to \infty.$2012-09-29
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    However i think that if we replace $n$ with factorial($n$) we do have a limit ?2012-09-29

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