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Define $f:\mathbb{N} \to \mathbb{R}$ by $f(n)=\frac{sin (\frac{n\pi}{4})}{n}.$

May I know if we can use L'hopital's rule to evaluate $\lim_{n \to 0} f(n)$ ? If not, how can we evaluate the limit without the use of series?

Thank you.

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  • $\begingroup$ The only sequences in $\mathbb{N}$ that converge to $0$ are the eventually constant ones. So $\lim_{n\to 0} f(n)$ exists if and only if $f(0)$ is defined (beforehand). $\endgroup$ – Daniel Fischer Jan 4 '14 at 12:10
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There is no such ting as $\lim_{n\to0}f(n)$ if $f$ is only defined on $\mathbb N$.

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An idea: for continuous variable $\;x\in\Bbb R\;$ , we have

$$\lim_{x\to 0}\frac{\sin\frac{x\pi}4}x=\lim_{x\to 0}\;\frac\pi4\frac{\sin\frac{x\pi}4}{\frac{x\pi}4}=\frac\pi4$$

Since the above limit exists in the above case, it also exists and equals the above for any sequence $\;a_n\xrightarrow[n\to\infty]{}0\;$ instead of $\;x\;$ ...

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