Quite sometime ago, I noticed something interesting regarding the saw-tooth function.

\[y=\begin{cases} \lim _{ n\rightarrow \infty }{ f^{ n }\left( x \right) -x } \quad \text{ n is an even integer} \\ \lim _{ n\rightarrow \infty }{- f^{ n }\left( x \right) -x } \quad \text{ n is an odd integer} \end{cases}\] \[f\left( x \right) =\sin \left( x \right) -x\] appears to describe the saw-tooth function.

I have no idea why this is so. If anybody knows, feel free to comment it. Thanks.

## Comments

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TopNewestYour limit is confusing to read. I'm guessing that what you want is

\[ y = \lim_{n \rightarrow \infty} ( -1)^n f^n (x) - x. \] – Calvin Lin Staff · 1 year, 10 months ago

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